Now, we need to make an estimate of the average force the steel ball
exerts on the panel when it is stopping. The force is the linear
momentum mv of the ball divided by the time the collision
lasts; I will assume that any hard ball will stop in about the same
time, so what matters is the momentum. It is easy to show that the speed
of the steel ball when it hits is v=4.5 m/s so mv=4.5
kg·m/s; so we need to find the size of a hail stone with a
momentum of 4.5 kg·m/s. But, we know the speed and mass from
above, so 4.5=(4.2x103R3)√(4.1x104R)=√(7.2x1011R7);
solving, R=0.031 m, m=0.13 kg, vt=36
m/s. So my best estimate would say that golf-ball sized hail would not
break your panel but hail larger than a golf ball by 50% or more would.
Keep in mind that all these calculations are estimates, not precise
calculations.
ANSWER:
Calculations of air drag are always an approximation. And the equations you are trying to use are certainly not going to be valid under such extreme situations—huge speed and tiny density. Furthermore, when these protons collide with the ship, they will very likely penetrate into the metal unlike the conditions for which equations like you are using are applicable where the gas molecules just bounce off the object. This will cause radiation damage to the ship's hull. Furthermore, at such high speeds interaction with photons will become important, in particular the cosmic microwave background, causing additional drag.
Your relativistic drag equation (which I do not where it came from)
cannot be dimensionally correct.)
QUESTION:
I am trying to figure out the actual force on a RV or Tinyhouse when traveling against a head wind. From what I recall it is not as simple as taking the vehicle speed plus head wind. For example, lets assume you are towing a tiny house at 60 mph against a head wind of 40 mph - one would think the total wind speed/force on the tiny house to be 100 mph.
However, I recall reading that the wind force increases as the speeds of both the trailer (tiny house in this case) and the head wind increase. E.g. 60 mph + 40 mph = 105 mph force but 60 mph + 50 mph head wind could have a force of 120+ mph, not 110.
I could not find info online for this. Is this true and if so is there a formula to work this out?
In the end I am trying to figure out if tiny houses should be built for hurricane force winds. If in fact, the above is correct and towing a tiny house in a head wind can be likened to a tiny house sustaining hurricane winds.
ANSWER:
Calculating air drag forces is an approximate exercise at best. For the
speeds you are talking about, the approximation that the force is
proportional to the square of the speed of the air is probably accurate
enough for your purposes, F∝v2.
There is no such thing as "mph force". There is no difference, as far as
air drag is concerned, between having a ground speed (GS) of 60 mph and
a head wind (HW) of 40 mph; and being at rest with a HW of 100 mph; and
having a GS of 100 mph in still air. I think what you are getting at is
that the force does not increase linearly with air speed; so if the
speed increases from 100 mph to 110 mph (an increase by a factor of
1.1), the force increases by a factor of (110/100)2=1.21.
That said, I can now give you a way to estimate the force (very
approximately) on the tiny house. Let A be the area which the
house presents to the wind; then F≈¼Av2,
which only works for SI units (meters, kilograms, seconds). So, for
example, if v=100 mph=44.7 m/s and A=(4 m)2=16
m2, F≈¼x44.72x16≈8000
N≈1800 lb. This would correspond to a pressure of about 0.07 PSI.
(A complicating factor is the presence of the towing vehicle. To some
degree, the tiny house would be shielded from the oncoming wind by the
vehicle. I see no way to estimate this effect because it would be
dependent on the vehicle.)
QUESTION:
What is the weight of impact (kgs) when a 0.5kg handsaw falls 11 floors at a speed on impact of 57mph?
ANSWER:
There is no such thing as "weight of impact". If you mean what is the
force which the saw exerts on whatever it lands on, you need to know the
time it takes it to stop, t. In that case, the average force
F during that time, it is
given by F=mv/t. We
need to convert mph to m/s, 57 mph=25.5 m/s. For example, if t=0.1
s, F=0.5x25.5/0.1=127.5 N;
the mass of a 127.5 weight is 127.5 N/9.8 m/s2=13 kg.
QUESTION:
Please explain me that why if a thin layer of water is spilled on a rough surface like plastered floor and we place our finger in it then why water move away from the point of contact of our finger on that surface and it appear to be dried around.
ANSWER:
This
is probably akin to the similar phenomenon of a foot pressing down on
wet sand and the area close to the foot is visibly dried. I found an
explanation on
Physics Forums which seems to be correct:
"The phenomenon being described is called 'Dilatancy' and was discovered by Reynolds about 100 years ago. It works only when you have well compacted sand that contains just enough water to cover all the individual grains. When you stand on the sand you create a stress / force which causes the sand to move. In order for the sand to move / flow individual grains have to be able to move past one another. Imagine a bunch of oranges stacked as you might see them at a grocers. The first layer has them all tightly arranged and then the second layer sits down into the gaps between the oranges on the first and third layer. Now imagine trying to move one of the oranges in the second layer. In order to move it the oranges on the first and third layer mut move down and up respectively to enable the orange to move. Effectively the volume of the pile of oranges or grains of sand increases with bigger gaps in between. So when you put your foot down on the sand it shoves sand out the way but in doing so the volume in between grains has to increase temporarily to allow the grains to move relative to one another. Consequently all the fluid at the surface is sucked by surface tension into the extra gaps made by the rearrangement of the sand. Since there is no longer any fluid at the surface the grains of sand are now dry."
If you go to the original Physics Forums question, ignore all the
early answers which are wrong.
QUESTION:
In magnetism, how can the magnetic field be used to trap charges inside a magnetic bottle? as this requires deceleration of the moving charge at either ends of the bottle and changing its direction of movement, this will need a work done by the magnetic force although the force will always be perpendicular to the velocity vector of the moving charge from Biot-Savart Law.
Please, if there is an image for clarification of directions of magnetic force and velocity vectors for more explanation.
ANSWER:
In the diagram, note that the forces on the charge are always toward the
center of the bottle. A full explanation can be found at
Physics Stack Exchange.
QUESTION:
If the law of conservation of mass states that matter can't be created or destroyed, then how is it possible in the deep future, after black holes die etc., the universe can be left with literal nothingness?
ANSWER:
There is no such thing as the law of conservation of mass. Mass
can be created or destroyed. In chemistry, conservation of mass is assumed in chemical reactions because the mass changes are so small as to be almost unobservable.
QUESTION:
Some people like to think about what would happen if the universe's constants were changed (see fine-tuning argument for the existence of God). But what if the laws of nature were themselves altered instead? What if, instead of E = mc^2, we had E = mc^3 or E = mc^4? Or what if the law of conservation of mass-energy or the equations governing electromagnetism were altered? Or what if instead of F = ma, we had F = m/a or F = 2ma?
ANSWER:
Your first example, changing the power to which c is raised in
the famous equation E=mc2, is impossible because for
any power other than 2 the equation is not dimensionally correct—e.g.,
mc3 does not have the units of energy; it would be
like saying "the speed of my car is 55 pounds/foot". Your second example
has two errors. F=m/a could be written as a=m/F
which would mean that the harder you push on something, the less it
would speed up, in violation of what happens. F=2ma would be
perfectly ok but would imply that you have defined what you mean by
force differently. Newton's second law is actually that the acceleration
is proportional to the applied force and inversely proportional to the
mass, a∝F/m or a=kF/m, where
k
is a proportionality constant. Assuming you have already defined what
mass (kg), length (m), and time (s) are, your choice of k
determines what you mean by force. The standard choice is k=1
which means that the unit of force (which we call 1 Newton) is that
force which causes a 1 kg object to have an acceleration of 1 m/s2.
If you were to choose k=½ (your F=2ma), one
unit of force would be that force which caused a 1 kg object to have an
acceleration of ½ m/s2.
You need to keep in mind that laws of physics are simply expressions of
how the universe works. For example, Coulomb's law simply states
that the force between two charges is inversely proportional to the
square of the distance between them, F∝1/r2
because this is what we find doing the most accurate experiments that we
can. But, we can not really be sure that the "true" law is not F∝1/r2.000000001
until we are able to perform an experiment which rules it out.
QUESTION:
I have two magnets, North poles facing each other (repulsive) Magnet A moves in the positive X direction, Magnet B moves in the negative X direction so that total momentum of system is zero.
The magnets come to a stop do to the repulsion of the same N poles. When they come to a stop, I "pin" them in place so they cant move. Total momentum was zero to begin with, both magnets stopped, total momentum is still zero. The kinetic energy of both magnets rushing towards each other, now stopped, the energy is now in the field between the magnets, if I was to weigh this system, with a sensitive enough scale, I would find the total mass of the system to be: Magnet A + Magnet B + Field so far so good?
Ok, I "unpin" the magnets, they fly apart with equal but opposite momentum, their kinetic energy coming from the energy stored in the field as mass. Is it wrong to say that the energy in the field is transferred directly to the magnets? or, is it more correct to say that "virtual photons" mediated the transfer of energy from the field to the magnets?
ANSWER:
This is a very difficult question even though it looks simple. First, I
think it is a mistake to say that the field has mass. The field has an
energy density U=B2/(2μ0)+E2(ε0/2);
yes, there is an electric field because the magnetic field is changing
in your example. As the two magnets approach each other the fields
change and therefore the energy content of the fields changes also. To
avoid the electrodynamics implied by your question (time varying fields)
and the resulting radiation (carrying energy away) which would occur, I
think we can get to the crux of your question by starting the two
magnets at rest and bringing them together by doing a certain amount of
work W and having them end at rest. If you measure the mass of
the whole system before moving the magnets to be M, its mass
after moving will be M+W/c2. Again,
thinking of the field as having mass is not the right way to think about
it, you need to just think about the mass of the system. For example, if
a nucleus has N neutrons and Z protons, its mass is less than Nmn+Zmp;
you would not want to say that the field holding the nucleus together
had negative mass, would you? Your second question, I think you can see,
is not simple either. Because the magnets are moving, there will be an
induced electric field. Because the magnets are accelerated, there will
be electromagnetic radiation carrying energy out of the system. I think
it is best to be thinking always about the whole system and not where
the energy "resides".
QUESTION:
I need to calculate the effort to hold up ONE end of an 80 pound beam to a 40 degree angle. What is the formula?
ANSWER:
I am not sure what you mean by "…the effort to hold up…" I
will assume you mean the force F (see figure)
you need to exert to hold the beam at rest. In general, it depends on
where the center of gravity of the beam is, how you exert the force, and
how long the beam is. The smallest force you would need to exert is a
force straight up (this statement is
incorrect, see below), so I will assume that is the case. If you analyze the
equilibrium problem for the beam in the figure, you find that F=Wd/L.
Note that it does not depend on the angle at all. For your case, if the
beam is uniform, i.e. d=L/2, then F=W/2=40 lb.
FOLLOWUP QUESTION:
Re: latest answer. The force required to hold up the end of the beam must involve a cosine of the angle.
ANSWER:
Details: Sum of vertical forces: ∑Fy=0=F+N-W;
sum of torques about point touching the ground: ∑τ=0=Wdcosθ-FLcosθ=Wd-FL.
Solutions: F=Wd/L and N=W(1-(d/L)).
This results from choosing to have F be vertical. I could have
chosen to have F perpendicular to the beam in which case the
answer would depend on θ. (I now see that my
statement in the original answer that a vertical force would be the smallest is
wrong, although my answer for a vertical F was correct and
independent of θ.) The new torque equation would be ∑τ=0=Wdcosθ-FL
so F=Wdcosθ/L and the 80
lb uniform beam at 400 would require a force of F=½(80)(0.77)=30.6
lb. It would require no force to hold the beam vertical since cos900=0.
QUESTION:
If gravity has the same effect as acceleration (according to the theory of relativity ) gravity curves light, does acceleration also cause a curve ?
And if it why do light particles move in a straight line ?
ANSWER:
The operative principle here is the equivalence principle: there is no
experiment you can do to distinguish between being in a uniform
gravitational field with associated acceleration g and
having an acceleration g in zero gravitational field. Suppose
that you are in an accelerating elevator in empty space which has a
small hole in the side. Someone shines a beam of light into the hole.
That beam will follow a parabolic path as seen by you inside your
elevator. Therefore, light will also be bent by a gravitational field.
There are two answers to your last question. First, if you watch the
photons from your Euclidian frame of reference, the photons do not
follow a straight line in that space. On the other hand, what mass does
is warp the space around it so a photon will follow the shortest path
between two points which you would call a straight line in that space
which is, itself, curved.
QUESTION:
So, I am learning about the basics of waves in class and I was bored so I started messing around with some equations.
λ=h/p p=mv, in the case of light, p=mc, c=λf E=hf c/f = λ c/f = h/p c/f = h/mc h=E/f c/f = E/mcf c=E/mc
E=mc^2
Is this a correct derivation of E=mc2 ??
ANSWER:
Well, "messing around" with equations you do not really understand
seldom leads to good results! First, E=mc2 does not
apply to light (photons) because light has no mass and you would
therefore you would conclude that light carries no energy which would be
nonsense; for more detail, see the
faq page. Your
major error, though, is right at the start writing p=mv which
cannot be true for a photon since it has momentum but does not have
mass. In fact, this is not even true for particles with mass since, in
relativity, the momentum is given by p=mv/√[1-(v2/c2)].
Incidentally, the energy of a photon is E=hf and the momentum
is p=h/c.
QUESTION:
a light photon moving at the speed of light, can it have spin? and if so can the spin be added to its speed giving it a faster than light speed breaking the laws of physics? I have always wondered this but never seem to find out if they have spin....
ANSWER:
Every photon has a spin of 1; this means that it has an intrinsic spin
angular momentum of L=√[1(1+1)]ℏ.
So, I am guessing that you are visualizing the photon as a little
spinning ball and think that you can conclude (see my figure) that the
"equator" on the near side of the ball is moving forward with a speed of
c+v. However, spin in quantum mechanics cannot be visualized using
such a simple classical model, even though we often do think of spin
this way to get a qualitative feel for spin. For example, if
you visualize an electron as a spinning uniform sphere with a reasonable
radius, you find that the surface must have a speed greater than the
speed of light.
QUESTION:
Say I fill an airtight barrel with water and have a valve at the bottom and a feeder hose at the top.
If this barrel is uphill and I have the feeder hose down lower say in a pond will the draining of the barrel through the lower valve create enough vacuum to pull the water uphill creating a siphon?
ANSWER:
First of all, I would call what you are proposing a pump, not a siphon.
You are trying to "suck" water uphill using a vacuum. The first thing
that comes to mind is that there will be a limit on how high the hill is
above the water level below. Even if you have a perfect vacuum, the
highest you can lift water this way is 10.3 m=33.9 ft. But you start off
with a hose full of air, so you will never get a vacuum, so you will be
limited further in the height to which you can pull the water from
below. For example, if the volume of the air in the hose were 1/10 of
the volume of the barrel, you could only lift the water 9.3 m. Or, if
the volume of the air in the hose were equal to the volume of the
barrel, you could only lift the water 5.1 m. So there is no simple
answer to your question, but this is probably not a very workable way to
lift water.
QUESTION:
If a person were in a closed container filled with water and the container was accelerated at high speeds, would the person in the container feel the g-forces the same as if they were not in the container?
ANSWER:
You would move backward until you hit the back wall and then the back
wall would exert a force forward on you. If there were no water, the
back wall would exert a force F1=ma
on you where m is your mass. If there were water, the
water would exert a force F2 toward the back on you
and the wall would exert a force F3 forward on you.
So now, F3-F2=ma so the
force on you from the wall would be bigger (F2+ma)
than if there were no water. Plus the water would be trying to
crush you.
QUESTION
ABOUT THIS ANSWER:
You recently answered a question regarding the effect of acceleration on a person in a closed container of water. You suggested the result would be movement to the back of the container and an increase in the force experienced by the person.
Buoyancy is dependent on relative densities, so a person will float with the same percentage immersed regardless of the local gravity/acceleration. This implies that the victim would, at least at moderate levels of acceleration, be forced to the front of the container.
Initially I thought that if the container was not full, it would be quite a comfortable experience since the accelerating force would be evenly distributed. However, humans are not of uniform density, so the persons chest with its air-filled lungs would be forced to the front and his bony, less buoyant extremities would be dragged the the back. Unfortunately, the questioner filled the container completely so the person would be pressed uncomfortably against the front wall.
At higher acceleration the body would be compressed sufficiently that he would become denser than the water, only then would he move to the back of the container to be further crushed by the water column.
ANSWER:
You are right, the motion depends on the density of the astronaut
relative to the density of the water. If the ship is in empty space (no
gravity) and not accelerating, there would be no buoyant force in any
direction and the astronaut would float either at rest or at constant
velocity until he hit something. The equivalence principle says that
there is no experiment you can do to distinguish between being in a
uniform gravitational field with associated acceleration g and
having an acceleration g in zero gravitational field. If the
ship had an acceleration a forward it would be the same as being in a
gravitational field pointing backward in the ship. Therefore, there
would be a buoyant force which would cause objects with smaller density
than water to move forward ("float") and objects with larger density
than water to move backward ("sink"); you correctly point that out and
in my original answer I was assuming an astronaut whose overall density
is larger (quite possible if he were wearing a heavy space suit, for
example). In the back wall case, my original answer was correct. In the
front wall case, the water would be pushing you forward and the wall
backward, so F2-F3=ma
or F2=F3+ma; now the
water pushes on you with a force greater than the wall pushes back.
QUESTION:
Most physicist say that an antimatter engine is impossible because there is no way to store the antimatter but if you were able to get raw positrons and electrons and store them in seperate electrically charged tanks (each with the same charge as the particle they are holding). would you be able to sucessfully store the matter and antimatter and decide when to open it and close it. So basically is the atraction between matter and antimatter stronger than the electroweak force?
ANSWER:
If you have a hollow conducting tank and charge it up with electrical
charge, the field inside is zero, so your plan to store charge inside
will definitely not work; the only force felt by the charges inside will
be the forces between those charges which will have the effect of
pushing them all out to the tank. All the positrons in the positron tank
would annihilate with electrons in the tank; all the electrons in the
electron tank would end up on the outer surface of the tank. And, by the
way, the only force between an electron and a positron is electrostatic.
QUESTION:
In the part of electromotive force, when we connect a wire to the terminals of the source of EMF like for example a battery, the potential difference between the two terminals create an electric field inside the conductor.
Haw fast this field been established, is it simultaneous just at the time of closing the circuit or it takes time even if it is extremely small amount of time?
ANSWER:
In a vacuum, an electric field propogates at the speed of light. In the wire
it will be a bit slower, but still close to light speed. The electric
field causes conduction electrons to move, hence creating the current.
QUESTION:
I have been studying atoms and strong and the electroweak force for about a week and i came across something that caught my attention. When a radioactive isotope undergoes beta decay one of its neutrons ejects an electron and an antineutrino and leaves behind a proton. my question is if it leaves behind a proton and jettisions an electron is there any chance that the nucleus with one extra proton than electrons in orbit around the nucleus could catch the loose electron?
ANSWER:
Why not just ask if you can combine an electron and a proton to make a
neutron? The answer is yes, and it is called inverse beta decay.
If a proton absorbs an electron, an electron neutrino will be ejected:
p++e-→n0+νe.
This most commonly happens when an atomic electron is captured into the
nucleus, a process called electron capture. This process also results
in the emission of an x-ray because the hole in the K-shell is filled by
a higher-orbital electron. It is also the main way
neutron
stars are formed.
Q&A OF THE WEEK, 3/19-25/2017
QUESTION:
Having a bit of a debate about whether this tennis ball would've landed in with a tennis player and we have a $100 bet on it. The ball machine fed the ball from the other side of the court at the baseline the player that hit the ball is a top ranked junior player...
the ball hit the ball machine edge 5 inches off the ground at the top of the wheel base and the player claims that it would've landed on the line if it had not hit the ball machine of which the picture demonstrates the point of impact is 5 inches above ground at the back edge of the line and there are no external elements such as wind as we are playing indoor.
the ball was traveling at approximately 30 mph at time of impact. We have attached an image for reference. We appreciate any clarity you could provide :-)
[Note that the angle relative to the horizontal is specified to be 600-700
in the photograph attached by the questioner.]
ANSWER:
My first reaction was to say that, of course, it would not hit the line.
That was because, as physicists often do, I was thinking of the ball as
a point and ignoring its size. You can see from the figure that there
could easily be a combination of h and v0
where the ball would strike the line had the obstruction not been there.
As best as I could tell, some part of the ball must touch the line so if
we calculate where the bottom-most point of the ball strikes the ground
(y=0) the ball will be in if x>0 in my coordinate
system. My guess is that if we simply assumed that the bottom point went
in a straight line in the direction of its initial velocity we would get
the right answer; but the ball is actually a projectile and moves in a
parabolic path so, since this is a $100 bet, I better do it right! The
equations of motion are
x(t)=x0+v0xt
y(t)=y0+v0yt-½gt2
where t is the time it takes to hit the ground and g=9.8
m/s2 is the acceleration due to gravity. Being a scientist, I
prefer to work in SI units, so I will do that. OK, let's summarize
everything we know. I will assume θ=700 since
that gives the best chance of hitting.
x0=R=2.7"=0.0686 m
y0=h-R=5"-2.7"=2.3"=0.0584 m
v0x=-v0cosθ=-30cos700
mph=-4.583 m/s
v0y=-v0sinθ=-12.59
m/s
y(t)=0.
So now the task is to put these into the equations above, solve the
y equation for t and put that value of t into the
x equation to find x. I find that t=0.00463 s
and x=0.047 m=1.9". Because x is positive, the ball
will hit the line. Going through the same procedure for θ=600,
I find t=0.00502 s and x=1.4", again hitting the line.
When the bet is settled, don't forget to
reward The
Physicist!
ADDED
THOUGHT:
The curvature of the parabolic path is, as I had speculated, miniscule.
For θ=700 above I found x=0.0474
m and if you just assume the ball went in a straight line to the ground
you find that x=0.0473 m.
FOLLOWUP QUESTION:
For clarity sake I am sending you two more pictures from the exact set up of which we have not moved.
Previously you only were sent the Sideview so I have included below sent a top view showing that the ball machine is actually out and the court view from the approximate angle that the ball was struck showing the approximate ballpark at the ball was hit from approximately contact was made 3 feet off of the ground.
I still can't wrap my head around if an apparatus is actually out and a ball strikes it from 5 inches above the court going in the opposite direction that it can still be in unless air resistance is involved of which there are no air elements. Just want to make sure that you still think the ball Woodland in from more data provided before I pay $100?
ANSWER:
The top view is helpful in verifying what I had assumed in the original
answer—the surface which is hit is aligned with the outer
edge of the line. The best way that I can convince you that it is
possible that my first answer is correct is to show two figures, one
showing a trajectory of the ball which lands it in bounds and one out:
Each figure shows the ball at the instant of impact with the machine and
the instant of impact with the floor had the machine not been there. The
dashed red line shows the trajectory of the center of the ball and the
solid red line shows the trajectory of the bottom of the ball. The
diameter of a tennis ball is 5.4". The ball with the steeper trajectory
(which would include your 600-700 trajectory) is
clearly in bounds by standard tennis rules.
FOLLOWUP QUESTION:
We did not know how to define the arc so we just said 60 70 as an ignorant guesstimate but we are ok with the actual real representation of the arc that we sent you in the last image...we just want to understand based on the ACTUAL real flight path arc that the ball really took when struck and its possibility of landing on the line of which the image with the actual art provided is a more accurate visual representation... and we are not sure of how to define the angle of approach based on that visual.
Our original explanation of the ball angle degree of approach to the baseline could have been lost in translation somewhat we are not certain... that's why we contacted you and that's why I sent a VISUAL representation because we really aren't sure of how to define that...we reckon the more information we provide you the more accurately u can clarify our understanding.
In saying that.... what approximate degree is the arc approaching the baseline from the image we provided this morning... assuming the ball was struck at about 3 feet off the ground traveling approximately 3 4 feet over the net? The distance from the net to the baseline is 39 feet and the player was approx 3 feet behind the baseline at time of contact so that would be approx 42 feet from the net at the time the contact was made. Does this change whether or not the ball lands on the line or is it irrelevant?
ANSWER:
This Q&A is turning into quite a tome! With the information you sent, I
can calculate a pretty good estimate of the trajectory of the ball
ignoring air drag. It turns out that your guess of 600-700
for the trajectory angle was way off. The algebra is tedious, so I will
just give the final results. To check that I made no algebra errors, I
plotted the trajectory, shown in the figure. I have worked in meters.
The ball is hit at x=0 from a height of 1 m, passes 1 m above
the 1 m high net at x=13 m, and then hits the machine at x=25
m just above the ground (y=0 m). You can see that my result
describes the path you specified quite well. Do not be deceived by the
picture, though, because the scales of the two axes are very different,
only 2.5 m for vertical motion compared to horizontal motion of 25 m, a
factor of 10. The inset shows the trajectory as it would look to the
eye. As you can see, the angle is much smaller than you estimated. The
analytical solution is that θ=15.60 and t=1.1 s; your estimate of the initial speed was pretty
good, 23.5 m/s=52.6 mph but the final speed (although it is not
important) is 23.9 m/s=53.5 mph not 30 mph.
Now I am compelled to recalculate with the best possible numbers whether
the ball hits in bounds or not. However, given all that I learned above,
I can make it brief: there will be a critical angle
θc=tan-1(y0/x0)=tan-1(2.3/2.7)=40.40;
any angle smaller than θc will be out of
bounds. For 15.60, x=2.7-(2.3/tan15.6)=-5.5", 5.5
inches out of bounds.
Finally, just for completeness, I would like to estimate the effect of
air drag. Using the approximation in an
earlier answer I find that t=1.13
s (0.03 s longer) and v=22.1 m/s (1.8 m/s slower). This would
correspond to the angle being a bit bigger, about 170, but not nearly
enough to cause the ball to drop in bounds.
Q&A OF THE WEEK, 3/12-18/2017
QUESTION:
How much lateral force is needed to damage bearings in the hub
motors of a skateboard when carving? So...if you roll in a straight
line, on a skateboard, you exert a radial load on the eight bearings
contained in the four wheels. But skating is more fun when you ride in
wavy lines - carving! So the bearings start getting a lateral or axial
load. How "hard" would the carve have to be (let's assume a rider of 100kg) to
break the weakest of the bearings? The
rotor is supported by two bearings, one that has an radial maximum force of 3.5
kN and the other of 7 kN. So the maximum axial forces would half those.
(The questioner and I had several
exchanges. The bearings are cylindrical and he was only guessing that
the axial force was half the radial force based on data for similar
bearings. I have edited his several emails to get the gist of things in
the question above.)
ANSWER:
The way I see the problem is shown to the right. The forces on the
skater plus skateboard are his weight mg vertically down, the
normal forces on the inner (N2) and outer (N1)
wheels, and the corresponding frictional forces f1
and f2. Whatever the rated axial (along the wheel
axis) maximum force is, that is what we want to use as the to find the
limiting conditions for the "carve"; the determining factors will be the
mass m of the skater plus board, the speed v he is
going, and the radius R of the path. For the lean, d
and h are the distances, respectively, of the center of mass
horizontally and vertically from the front wheels; note tanθ=d/h.
The wheel base is s. The normal forces and frictional forces
shown each represent the forces on two wheels. Note that the inner
wheels carry the most force.
The easiest way to do this problem, since it is an accelerating system,
is to introduce a fititious centrifugal force C=mv2/R
pointing outward. Newton's equations are N1+N2-mg=0
(vertical forces), f1+f2-C=0
(horizontal forces), and N1s+mgd-Ch=0
(torque about inner wheels). Now, from the torque equation, there will
be a maximum speed you can go for a given m and R
before the outer wheels leave the ground. At this time N1=0=(Ch-mgd)/s
or v2/(gR)=d/h=tanθ.
Also, f1=0, so f2=C=mv2/R.
Now, the inner two wheels each are experiencing the axial force of F=½mv2/R.
Just to do an example, let v=10 mph=4.47 m/s, R=5 m,
and m=100 kg; then F=0.2 kN. If your guess that Fmax≈1.75
kN is correct, you should be ok. For this example, θ=tan-1[v2/(gR)]=220. If you execute the turn with
a smaller lean angle, all four wheels will share part of the load. To do
the general solution where both wheels have N≠0 would
not be two difficult but would be quite a bit messier algebraically and
probably not all that useful.
FOLLOWUP QUESTION:
Thanks for the explanation on your website. It wasn't really what I was looking for because it doesn't really give the answer something layman can understand.
Also, you compare your estimated lateral force of 1.96 kN
(0.2 kN, see below) to 1.75 kN, although they would be the breaking limit for only one bearing.
You assumed that just the 2 inner wheels receive the load, so isn't that a total of 4 bearings... shouldn't the breaking limit be 4*1.75?
ANSWER:
I am glad you asked this question because it got me looking more
carefully at what I had done late last night and I found an extraneous
factor of g had crept into the final stage of my calculation, ("…so f2=C=mgv2/R…")
so my final answer was too big by a factor of 9.8 meaning that the
lateral force per wheel is F=0.2 kN; this has been corrected in the
original answer. So you should have no problem after all. I was in the
process of writing an email trying to "verbalize" my answer somewhat to
make it more accessible when I discovered this. Here is what I wrote:
You asked me "How ‘hard’ would the carve have to be (let's assume a rider of 100kg) to break the weakest of the bearings?" That is what I gave you. I assumed that the weaker of two bearings in a wheel can break even if the stronger does not.
(I assume they are coaxial, one inside the other.) Sorry if the explanation was too technical, but that is as basic as I can get to convey the details. A few comments to try to clarify:
-
What I call
C, the centrifugal force, determines how "hard" the carve is.
C=mv2/R. For the example I did, C=100x4.472/5=400 N.
-
When carving, the harder you carve the greater the load will be carried by the inner wheels on each axel. Eventually, the outer wheels will just lift off the ground which necessarily results in the inner wheels taking all the load, both lateral and radial.
Since you asked me for how to calculate the maximum lateral force
any wheel would experience for a given m, v, and
R, that is what I gave you.
-
I do not
understand your last question but I do know that if two wheels
experience a force of 0.4 N then each experiences a force of 0.2 N
(assuming they equally share the load).
ADDED
THOUGHT:
I guess I have not really fully answered your question
"How 'hard' would the carve have to
be…to break the weakest of the bearings?" I suggest that the
appropriate equation would be 1.75x103=½mv2/R.
Here are a couple of examples:
-
What is
the fastest a 100 kg rider could go in a curve of radius 10 m? v=√(2x10x1.75x103/100)=18.7
m/s=67 km/hr=42 mph.
-
What is
the tightest curve a 100 kg rider could turn at 30 mph=48 km/hr=13.4
m/s? R=½x13.42x100/1.75x103=5.1
m.
These, of
course, assume 1.75 kN is the strength of the weakest bering.
QUESTION:
In a black hole what becomes of the system's angular momentum?
ANSWER:
Black holes have angular momentum. Any object which has angular
momentum, will add its angular momentum to the black hole's when it is
captured, conserving angular momentum.
QUESTION:
If the earth is spinning at 1040mph to the EAST, why do trips from LA to NY, take the same amount of time in either direction, give or take 30min?
If the plane is traveling at 500mph, wouldn't flight times be significantly different from LA to NY vs NY to LA?
ANSWER:
The atmosphere rotates with the earth, to first approximation. The
airplane flies relative to the air and therefore takes the same time in
either direction because it is flying in still air. In reality, at the
altitudes where commercial airliners fly, there is a strong air current
called the jet stream which moves, relative to the ground, in a west to
east direction. Therefore the trip from NY to LA is usually longer.
QUESTION:
Since a perfect black body would absorb all electromagnetic radiation would it mean that it would also have a perceived temperature of absolute zero and wouldn't it also mean that eventually the temperature of whatever the material the black body itself is made up of would eventually reach the planck temp since it's always absorbing radiation and never releasing it?
ANSWER:
A black body is also a radiator. If it is in a radiation field it will
absorb all radiation striking it and, as it absorbs this energy, it will
increase in temperature. As it gets hotter it will increase its own
radiation. Eventually it will come into thermal equilibrium with its
environment, absorbing and radiating at the same rates.
QUESTION:
Good Day ! Thank you for being there and dedicating the time to answer our questions. I have many... Can I start with the "Inverse Square Law" as it pertains to light. E = I/r2 , Using 100 as "S", S/4pi2 = I ( intensity at surface of sphere ) as the source strength, the farthest we would be able to "see" would not even get us to PC. it is 4.2 light years away, 5,878,499,817 x 4.2 = 24,689,699,231.40 miles... Assuming 100% at "S". Reaching the surface of the earth, using a nominal, 8,000 mile diameter, that works out to be about 1 (one) photon for every 1,205. square miles.... And, that is purported to be our closest star. How can this be ?
ANSWER:
I generally do not try to find errors in questioners' calculations, I
just do the calculation myself. The sun is a pretty average star and its
photon
rate is about Φ≈1045 photons per
second. Using R=4.2 ly=4x1016 m, I=Φ/(4πR2)=1045/(64πx1032)≈5x1010
photons/s/m2. That would be the photon intensity 4.2 ly from
a sun-like star.
QUESTION:
If you would like to send a space probe to an asteroid that is 7 AU away, how many years would the journey take using the minimum energy direct trajectory?
ANSWER:
You seem to think that energy must be expended to keep it moving. In
fact, once you give it a velocity greater than the escape velocity, it
will coast the whole way. The minimum velocity you must give it is just
slightly less than the escape velocity, but for the purposes of this
answer, let's just say that we give it the escape velocity which is
about 104 m/s. I find that the speed at r=7 AU would
be about 28 m/s and the time to get there would be about 1.5x1012
s≈48,000 years. So the energy required would be that amount needed
to give your probe the escape velocity from earth's surface. Keep in
mind that this estimate ignores everything except the earth. It would be
a better question to ask, for example, how much energy would be needed
for the probe to arrive in 10 years.
QUESTION:
I'm hoping this is not considered off the wall but our moon does not have an atmosphere. And our moon and earth are spinning (quite fast) and also our solar system is spinning while traveling through space. so my question would be how those men weren't blown clear off the moon when they landed on the moon?
ANSWER:
Where did you get the idea that the moon is "spinning quite fast"? The
moon spins once on its axis approximately every 28 days so that it
always keeps the same side pointed toward the earth. The earth, of
course, rotates once every 24 hours. You can easily calculate the
centrifugal force on something on the equator of the moon or earth, that
force which I presume you are assuming "would blow [us] clear off". On
earth that force is less than ½% or your weight; on the
moon that force is 2.5x10-16% of your moon weight! So, you
see, in no way do you have to worry about being "blown clear off"!
QUESTION:
I am a Middle school science teacher and we do an unit on cell phones as part of our NGSS "waves" unit.
The performance task is to build a device that blocks a cell phone signal. The kids quickly figure out (from the unit lessons and the internet) that aluminum foil is the best solution.
My Question(s): What exactly is happening with the aluminum foil? everything I read says the waves are being absorbed, are they creating a Faraday cage, or just creating a barrier that blocks the waves?
I have had students create "Faraday Cages" (with copper wire, following internet plans) and it did not work. What exactly is a Faraday Cage and how does it work?
My teaching partner noticed a correlation between when the phone is touching the box, and when it is insulated from the box (with a cell phone case). Is there something to this?
This is the second year of this project and I am in awe at the opportunities for learning for myself and the kids, thanks for your time.
ANSWER:
Do your students know what an electromagnetic wave is composed of? There
are oscillating electric and magnetic fields as shown in the figure. The
important thing is that there are electric fields and electric fields
cannot penetrate into a hollow conductor (the Faraday cage). To
understand how the Faraday cage works, look at the animation. When an
electric field is applied the electrons experience a force opposite the
field and migrate to one side of the cage while the other side has a net
positive charge because of the missing electrons. Amazingly, the
electrons arrange themselves in just the right way that their electric
field is exactly opposite what the external electric field inside would
be, giving a net field of zero inside. The waves are not so much
absorbed as they are cancelled out. A Faraday cage does not have
to be completely closed if you want to keep radio waves out as long as
the size of the holes is very small compared to the wavelength of the
radiation; it could be made of chicken wire, for example, as long as the
wavelength is much greater than a couple of inches. From what I can find
out the frequencies are in the not too far from f=1 GHz=109
s-1; since the wavelength of a wave is the speed divided by
the frequency, λ=c/f=3x108/109=0.3
m. So, I would have thought that a cage you made would have gaps much
less than 30 cm. Were all the wires in good contact with all others?
Often copper wires have a varnish-like coating on them to serve as
insulation. I don't know what you mean by "correlation", so I cannot
answer that part of your question. (By the way, the aluminum foil is
a Farady box if it completely surrounds the phone.)
QUESTION:
How much heat would it take to heat 1 gallon of water to 600 deg F in a pressurized system, from 70 deg F to 600 F in 1 hour. Not counting the ss vessel.
Also since the water is not allowed to change states are the calculations just the Sensible heat cals or are there special calculations needed.
This is part of a R&D Application.
ANSWER:
For my approximation to be fairly accurate, the water must remain liquid
at a constant volume. I will work in SI units so 700F=210C
and 6000F=3150C; I will convert back to Imperial
units for the final answer. I looked up
data for the specific heat of water which turns out to have a
significant temperature dependence as shown in the figure (black). I did
a quadratic fit (green) to these data and integrated over the
temperature range to get E=1356 kJ/kg. The mass of a gallon of
water is about 3.8 kg so the total heat is Q=1356x3.8=5.2x103
kJ=1.44 kW⋅hr=1240 kilocalories. Keep in mind that the pressure
will be very large at 6000F, about 1800 psi.
QUESTION:
I have been reading a lot lately about the prospects of nuclear fusion reactors; especially regarding electrical power plants. My question is this: if the hot plasma from the reaction is shielded from the equipment to protect it from the heat by a magnetic bottle, how then do you access the heat to make electrical power via steam for turbine generators or direct conversion from heat to electricity.
ANSWER:
The purpose of the magnetic bottle is to keep the hot plasma isolated
from the physical containment; just touching the containment vessel
would present two problems—it would cause instabilities in
the plasma and it would damage the vessel. But the plasma is hot and
anything hot will radiate heat energy; in other words, the magnet bottle
contains the plasma, not the heat. This radiant heat would rapidly heat
up the vessel which would heat up some coolant mechanism (imagine having
water tubes coiled around the outside of the vessel) which would carry
away the heat to drive the turbines.
QUESTION:
We could induce artificial gravity through centripetal acceleration. For example a ring-like structure in a spaceship could rotate about 1.34 rpm if the radius to the centre is 500 meters. This will give 1 g at the edges of the ring.
However we can also induce artificial gravity in the spaceship through its propulsive power and hence a constant acceleration at 1 g of the spaceship is required.
But what if the spaceship is accelerated at more than 1 g in order to achieve 75% the speed of light, even though the spaceship consists of the 500m radius ring-like structure that rotates at 1.34 rpm. What type of artificial gravity will the occupants feel? Will the excessive acceleration cancel out the centripetal effects in the ring? Is there an equation or formula that combine both linear acceleration of an object while the objects is also rotating?
ANSWER:
You seem to think that you need an acceleration greater than
g to get to 0.75c, but any acceleration will do. But,
since you seem interested in a greater than g acceleration, I
will choose a0=1.5g. So now a person of
mass m in the ship sees two fictitious forces, mg
which is radially out and 1.5mg which is
toward the rear of the ship. A person in the ship would experience a net
force of about 1.8mg pointing out and back. This would not be a
comfortable situation. If you are just going to accelerate constantly, I
would recommend not messing with the rotation at all, set your
acceleration to g, and make all the "floors" in the ship
perpendicular to the acceleration. If you are interested in how long it
would take to reach 0.75c with a0=g,
you can read off the graph (which I took from an
earlier answer) that gt/c=1.8,
so t=5.5x107
s=1.7 yr.
QUESTION:
A worker was pushing a pallet forklift and when he put the metal against the metal at the bottom of a glass door - the glass shattered to pieces. He just barely touched the metal on the bottom of the door. What could have happened to make it shatter like that?
ANSWER:
Glass sometimes breaks under circumstances where it does not seem that
enough force has been applied. As you probably know, glass is formed in
very high temperatures and allowed to cool. Sometimes the cooling does
not occur uniformly over the whole volume of the glass and the finally
cooled object can have places where there are very large internal
stresses. Just a small force at such a location can then cause the glass
to break. I cannot verify that this was the case for your situation
because it is a little hard to imagine a forklift having "just barely
touched" anything!
QUESTION:
Is it true that the electric energy of one electron that is moved through a potential difference of 1 volt is called electron volt?
ANSWER:
You are on the right track, but your terminology is a little shaky (it
is not clear what "electric energy" means). There are two ways you might
want to define the electron volt (eV). First, if an electron at rest is
allowed to accelerate across a potential difference of 1 V, it acquires
a kinetic energy of 1 eV. Second, if you push an electron across a
potential difference of 1 V (opposite the direction it wants to go), you
will do 1 eV of work. 1 eV=1.6x10-19 J. You need to realize
that if the potential difference is greater than a few hundred volts,
the kinetic energy will not be exactly ½mv2
because of relativistic effects.
QUESTION:
why are sparks more likely to occur between two charged particles closer together rather than far.
ANSWER:
For air to become a conductor, there must be a sufficiently strong
electric field to ionize the air molecules. There must therefore be a
potential difference (voltage) between the two electrodes. The air has a
dielectric breakdown strength of about 30 kV/cm which means that 30,000
V are needed for a gap of 1 cm but only 3000 V is needed for a gap of
1mm.
QUESTION:
If I drove my car home with the boot lid open, why would the air resistance be higher than usual?
ANSWER:
It probably would be higher, but not necessarily. If it were higher, it
would be due to the fact that the area presented to the onrushing air
was larger and the drag is proportional to the area. On the other hand,
aerodynamics can be very nonintuitive. The spoiler on some cars is
designed to break up the smooth flow of air over the car which actually
results in lower drag at high speeds. The dimples on golf balls and the
hairs on tennis balls have the same purpose, to break up smooth air
flow.
Here is an anectdote which illustrates that your intuition is not always right regarding drag. Some years ago somebody called in to
Car Talk on NPR and asked about these nets you can buy to replace the tailgate in a pickup truck to reduce air drag. Makes sense, right? The tailgate is like a wall in the wind and to get rid of it will reduce drag and increase your mileage. Click and Clack said that they thought these things were a great idea for reducing drag and increasing fuel efficiency. During the intervening week before the next show an engineer from GM called in and told them that removing the tailgate in fact greatly increases the overall drag on the truck. The reason is that the tailgate traps a bubble of air which rides along with the truck and the headwind slips over it. There was a lot of crow-eating that week at Car Talk Plaza!
QUESTION:
Does temperature affect a magnet's magnetism?
ANSWER:
Magnetism is a vast field and the temperature dependence depends on the
material. I will just address a simplified description of a simple
ferromagnet, a common permanent magnet. The basic way that these work is
that electrons interact with their neighbors in nearby atoms such that
their magnetic moments align with each other giving the bulk material a
net magnetization. The most important temperature dependence is that,
for any ferromagnetic material, there is a critical temperature called
the Curie temperature where the magnetization disappears. So, if you
want to destroy a permanent magnet, heat it up above the Curie
temperature; unless you cool it in an exteranal magnetic field, it will
not be a very good magnet when cooled. The behavior below the Curie
temperature is likely to be complicated, but there will be generally a
decrease in magnetization as it heats up.
QUESTION:
Photons don't have mass, but they have momentum. So this means that if a photon hit a mirror, for example, it pushes the mirror a little bit forward. Thus, the photon transferred momentum to the mirror. So this would mean that the photon lost a fraction of its speed or its mass decreased because p=m*v! But how could this be if photons are massless and only can travel at the speed of light?
ANSWER:
The linear momentum of a photon is not p=mv (since it could not
have any momentum if m=0), but rather p=hf/c
where h is Planck's constant, c is the speed of light,
and f is the frequency the corresponding light wave. In
everyday life, the light which reflects from a mirror is the same color
of light that went in; this means that the mirror did not recoil at all,
essentially has infinite mass. If you take into account the recoil of
the mirror, some of the photon's momentum would be transferred to the
mirror which would mean the photon would have to lose some momentum. But
h and c are both constants of nature, so f
would have to decrease; that would mean that the actual color of the
photon would be slightly shifted toward the red end of the spectrum. You
could never do this experiment, the shift is too tiny. But, if you
"reflect" a photon from an electron (shoot gamma rays or x-rays at a
solid which has many electrons in it) you can easily measure the change
in wavelength of the radiation; this is called
Compton
scattering and was one of the pivotal experiments in the development
of quantum physics.
QUESTION:
Is the terminal velocity of a matchstick falling from 1000 metres the same as a 3000lb car falling from the same height.
ANSWER:
The terminal velocity is the largest speed something will acquire if it
could fall forever. So it has nothing to do with how far it falls; a
feather may acquire 99.999% of its terminal velocity after falling one
inch but a bowling ball might have to fall a mile before it reached
99.999% of its terminal velocity. This problem you can do just using
your common sense (although I will do a rough estimate below). Something
as light as a matchstick will not fall far at all before it is moving
with nearly its terminal velocity; if you had a terminal velocity as low
as a match's, you could jump off the Empire State Building and not get
killed. The car would go much faster.
This may be all that you want, but for the interested I will roughly
calculate some numbers. The terminal velocity v of something
with mass m which presents an area A to the onrushing
air can be roughly approximated as v=2√(mg/A)
where g≈10 m/s2 is the acceleration due to
gravity. The match I estimate to have m≈0.1 grams=10-4
kg and area A≈(5 cm)x(1 mm)=5x10-5 m2;
then v≈9 m/s. The car I estimate to have m≈3000
lb≈1400 kg and area A≈(2 m)x(3 m)=6 m2;
then v≈97 m/s.
QUESTION:
I was playing with my nephew and his toy cars and came to notice that when pushed, after a short distance, the cars begin to veer off course. Some simple "testing" determined this to be a (seemingly) random effect. This was not really new to me, but curious nephew eyes then began to ask me "why this is so". I assume it to be a mix of many effects arising due to the (lacking) production quality of the cars, but I didn't find a really satisfying answer. Perhaps you could give a "best guess" as to which effect is the main contributor?
ANSWER:
Suppose there is some small force which pushes the car left or right at
a rate of 0.4 ft/s/s and there is some larger force which causes
it to slow down at the rate of 3 ft/s/s; now push the car so it has an
initial speed 10 ft/s. The path followed by the car until it stops will
resemble the path shown in the figure. That is my best guess!
QUESTION:
In space like sci fi dogfights (gundam or macross) is it possible to accelerate from a stationary (from the pilots perspective like a ship) object straight forward (thinking in 4 axis) then if accelerate directly down from your original vector then accelerate again in a new vector does the orignal speed change or do only the amount of gravitational energy change? Also what would it do to the person inside the craft? If your already doing 500mph and you change directions like I said down at another 500mph then again at 500mph does it add up or is it like launching from a craft already doing 500mph where the energy felt going foward is 0 from their perspective? Like when a astronaut inside the ISS goes out for a space walk? Do they feel the pressures when they wear those jetpacks (sorry for childish name) and change directions? Again what type of math would I study to learn the answer to these types of questions? and hopefully solve them some day because space has space for everyone and I want people to one day go to the stars ALIVE!
ANSWER:
I
am afraid that your question is quite muddled, but I think it shows that
you do not really understand what acceleration is. Average acceleration
is defined to be the change in your velocity (the difference between the
final and initial velocities)
divided by the time to affect the change. If I understand your example
you are originally moving with speed vinitial=500
mph in one direction and later are moving with speed vfinal=500
mph in a direction perpendicular to your original direction of travel;
it is important that you realize that velocity is a vector so that it
changes if the direction changes but the speed stays the same. The
difference between the two velocities is shown in the figure, Δv=vfinal-vinitial.
So the average acceleration is a=Δv/Δt
where Δt is the time to make the maneuver. The average
force F you will experience is in the
direction of a and proportional to your mass
m, F=ma. For
example, suppose your weight is 160 lb (then your mass is m=160/32=5 lb⋅s2/ft), Δv=500√2=707
mph=1037 ft/s, and Δt=1 s. The average force you will
feel is F=5x1037/1=5185 lb! You do not want to change your velocity too
quickly!
QUESTION:
I'm just curious: is there any geometric structure that could be built-up infinitely without collapsing under it's own gravity? A tetrahedral mesh, like diamond maybe?
ANSWER:
Let's refrain from talking about building up "infinitely" since
there is not an infinite amount of energy in the entire universe. The
gravitational field g of a point mass m
is g=-mG1r/r2
where G is the universal gravitational constant, 1r
is a unit vector pointing radially out and r is the distance
away from the point. To get the gravitational field for an object not a
point you need to divide it up into infinitesmal pieces, each having a
field dg=-dmG1r'/r'2
where r' is the vector from dm to the place where you
wish to calculate the field; you then integrate over the entire object,
usually not a trivial exercise. This is one of the main reasons that
Newton had to invent the calculus, so that he could prove that he could
treat the sun and the planets (spheres) as point masses. I understand
that the difficulty of proving that the field of a spherically symmetric
mass distribution outside the object is identical to the field there if
all the mass were in a point at the center caused a delay of something
like 20 years in the publication of his theory of gravity. I will only
talk about spherically symmetric masses here because, the point mass
field being spherically symmetric, all large bodies will tend toward a
spherical shape. This is why the stars and planets are spheres; see an
earlier answer on
cylindrical masses. Calculation inside the object is trickier: a uniform
sphere of radius R and mass M has zero field at the
center which increases linearly until the surface where it has magnitude
g=MG/R2; a hollow sphere of radius R
and mass M has zero field everywhere inside and then jumps
discontinuously to g=MG/R2 at the surface.
Since we know that a star with a large enough mass will collapse to a
neutron star and/or a black hole, nothing will stop that from happening,
certainly no "mesh" of any kind. On the other hand, if you want to
create a hollow sphere with a thin outer shell, I believe that there
would be no limit to how large you could make it without its collapsing.
Suppose you have a hollow sphere of radius R1 and
and mass M1 and it does not collapse; the field
pushing in at the surface is g1=GM1/R12.
Now suppose you have a hollow sphere of radius R2
and and mass M2=2M1; it is easy
to show that if you keep the surface density of the shell σ=M/(4πR2)
constant then R2=(√2)R1
and so g2=GM2/R22=2GM1/(2R12)=g1.
The field remains the same and so the field per unit area actually gets
smaller as the shell gets larger, making it even less prone to collapse.
QUESTION:
What's the physics of orbiting? Why don't satellites lose their velocity over time and fall straight to the earth's surface as they're orbiting (falling) around the planet?
ANSWER:
Orbital mechanics is just applications of Newton's laws of
motion: the motion is determined by all the forces on an object. Start
with the simplest assumptions: there are only to bodies and the mass of
one body is hugely greater than the other and the only force is gravity.
This is what is called the Kepler problem. See an
earlier answer for the several possibilities of orbits but the
important thing is that a stable orbit is the solution to Newton's
equations and it simply continues forever until some other force changes
it. Next you take into account the masses of both objects and find that
the solutions are essentially the same except that the two orbit around
their center of mass and the reduced mass μ=Mm/(M+m)
replaces the mass M in your equations. Now you can add other
non-ideal forces to the problem. For example, although earth satellites
are above most of the earth's atmosphere, they are not above all of it
and the air drag with the very thin atmosphere gradually takes energy
away from the satellite and its orbit eventually decays to where it hits
the ground. The farther out the satellite is, the slower this decay is.
Other things can cause the orbit to change also. For example, the moon
causes tides on the earth (in both the oceans and the solid earth) which
causes some of the energy lost by the earth to be gained by the moon so
it is gradually moving farther away (by like about 4 cm a year). And of
course any third body perturbs the nature of the orbit. For example, the
planet Neptune was discovered in 1846 because the observed orbit of
Uranus was not exactly what its Kepler orbit should have been because of
interaction with another body, Neptune; calculations predicted the
necessary orbit of the unknown planet and it was subsequently observed
where it was predicted to be. Imagine doing such a complicated
calculation long before computers.
QUESTION:
Would a stove-top pot heat liquid faster if the bottom surface was concave instead of flat? My theory is the increased surface area on the liquid in the pot would cause it to heat faster, and heat would travel 'up' the concave 'cone' area so it would not lose any heat (or proportianally not a lot) compared with if that area was flat and closer to the flame.
ANSWER:
Such a pot would definitely not be best for an electric range.
Here you want the element to be in contact with the metal pot bottom so
that conduction is the main way of getting heat into the pot; with your
pot convection would be the mechanism and result is smaller heat flow.
For a gas range, it is possible that some advantage could result as you
speculate, but I doubt it. The hottest point of a flame is in the
visible part of the flame that you see, and you want this to be close to
the pot.
QUESTION:
If traveling on an airplane from North Pole to Brazil, the pilot cannot fly on a straight line or he will miss his target; the pilot must actually fly on curved path to reach his desired destination. What is the reason for this?
ANSWER:
This is probably a homework question, forbidden here, but I am
going to answer it because of its ambiguity. The problem with this
question is that it does not define what is meant by a curved path. Of
course, since we live on the surface of a sphere, no path between any
two points on the surface of the sphere is a straight line in the three
dimensional space from which we might view that path. Only by digging a
tunnel between the two points is the path a straight line. But if you
travel through this straight tunnel and your path is observed from
outside the earth you follow a curved path because the earth is
rotating. How do we usually define a straight line? It is the shortest
path between two points. So if you observe things from our point of view
on earth and define a straight line on this two-dimensional surface to
be the shortest distance between two points, you can connect any two
points on earth by a straight line. If the pilot flies due south on the
longitude 43.20 W, he will end up in Rio de Janeiro. The
pilot does not have to keep steering his plane to account for the
earth's rotation (which is what is hinted at here, I think); he flies
relative to the air and the air rotates with the earth. If this is a
homework question, it is a pretty stupid one.
Q&A OF THE WEEK, 2/19-25/2017
QUESTION:
Where can I get a graph (or other information) about the increasing "relativistic resistance" to the acceleration of a particle (an electron, hopefully) as its velocity is increased to near-relativistic speeds? If such a graph is not available, then how can I calcuate this increasing force that resists a particle being accelerated when it is already traveling at some relatively high percentage of the speed of light, like maybe 70%?
ANSWER:
It would definitely behoove you to read an
earlier
answer first which has lots of details and discussion of
acceleration. There are two ways to approach this problem:
-
Suppose
the observer in the inertial frame is doing the pushing and wants to
know how hard to push the particle to achieve a particular
acceleration.
Resistance to acceleration is usually called inertial mass and the
inertial mass m of a particle with rest mass m0
and speed v is m=m0/√[1-(v/c)2].
The first figure above plots
m/m0 as a function of v/c.
So, to achieve an instanteous acceleration of a, a force of
F=ma=m0a/√[1-(v/c)2];
so for your example of v/c=0.7, you can read off
the graph that you would need to exert a force 1.4 times larger than
you would if the particle were moving slowly. This is probably what
you want if you are interested in electrons accelerating since you
would be accelerating them.
-
Suppose
the observer was on the particle and the particle was a rocket ship.
You adjust your engines so that the force F which they
exert on the ship causes a constant acceleration of a0=F/m
where m is the rest mass of the ship. What
acceleration a does another observer in an inertial frame
(on earth maybe) see when the rocket has a speed v?
Starting with the velocity derived in the
earlier
answer, v=(a0t)/√[1+(a0t/c)2],
we can calculate the acceleration by differentiating with respect to
t, [dv/dt]/a0=a/a0=(1+(a0t/c)2)-3/2.
Now, reading off the second graph, when v/c=0.7,
a=0.36a0; the stationary observer will
only see 36% of the acceleration seen on the ship.
ADDED
NOTE:
It is important to note that the acceleration observed depends
on the inertial frame you are in. In way#2 above, two different inertial
frames, say with v/c=0.7 and 0.9, will observe the
ship having different accelerations. This is why acceleration, and
therefore also force, in special relativity do not play an important
role as they do in Newtonian physics where all inertial observers see
the same acceleration. This is discussed in the
earlier
answer.
Q&A OF THE WEEK
(2/12-18/2017)
QUESTION:
If I am moving 55 MPH East (or West) at the equator how much weight would I gain (or lose) due to the Eötvös Effect. Thank You in advance. I am 73 years old and too dumb to figure this out myself.
ANSWER:
First of all, weight is the force the earth exerts on you so you
never gain or lose weight when you are moving; you might want to say
"apparent weight" which is the force which would be measured by a scale
you were standing on. You experience two real forces, your weight
W down
and the normal force N (a scale, for example) up. One way to solve this
problem is to note that an object with mass m with speed v
moving in a circle of radius R has an acceleration a=v2/R
which points toward the center of the circle; then apply Newton's second
law, F=ma=mv2/R=W-N and solve for
N to get your apparent weight of W-mv2/R,
smaller than your actual weight. This is the
"Eötvös effect".
But there is another way to approach the problem. Rather
than solving the problem from the outside the earth, we might want to
solve it here on the earth. But Newton's laws are not valid in an
accelerating reference frame (accelerating because it is rotating). You
can force Newton's second law to work, though, by inserting a fictitious
force which I will call E for
Eötvös but it is more
commonly known as the centrifugal force; E=mv2/R
pointing radially out. Newton's first law now applies, N+E-W=0,
so, again, N=W-mv2/R=W[1-v2/(gR)]
where g=32 ft/s2. In the figure above you have a velocity
v=vEarth+vman. If you are at
rest, v=vEarth=1040 mph=1525 ft/s and R=3959
mi=2.09x107 ft; so N=W(1-0.00348) and a scale will
read 0.348% smaller than your actual weight. If you move with a speed of
55 mph=81 ft/s in an east direction, v=1525+81=1606 ft/s and
N=W(1-0.00386), 0.386% smaller than your actual weight. If you
move with a speed of 55 mph=81 ft/s in a west direction, v=1525-81=1444
ft/s and N=W(1-0.00312), 0.312% smaller than your actual
weight.
FOLLOWUP QUESTION:
I made a donation, but the way I read your answer you have both directions being SMALLER, if I read it right. I know that West is an increase and East is a decrease, just thought you would like to know.
ANSWER:
Thanks for your support! Very generous, particularly because you
say that I am wrong! I must stand by my calculations, though. It is
certainly not unheard-of that I make an error, but this answer is right.
Since I define weight to be what a scale would read if the earth were
not rotating (or at north or south poles), you will see that the
apparent weight (what the scale reads) increases if you go west and
decreases if you go east, just the same as what you "know"! All apparent
weights are smaller than the actual weight; the only exception is if you
go west with speed of vEarth in which case the
actual and apparent weight will be the same. If you want to compare to
your apparent weight at rest, it is 0.386-0.348=0.038% lighter going
east, 0.348-0.312=0.036% heavier going west.
QUESTION:
How much "work" (physics definition) is actually accomplished in a gym workout?
I'm currently using F x D x reps = actual work done.
The upward lift ("D") x the amount of weight lifted ("F") x the number or repetitions... to get actual work done.
Am I in making some mistake, here?
Thanks, in advance, for your help.
ANSWER:
By "upward lift" I assume you mean the distance lifted. So,
lifting a weight F over a distance D you would do
W=FD units of work on the weight. For example, the weight of a 2 kg
mass is about 19.6 N and the work to lift it 1 m is 19.6 J. But, and
here is the catch, you use more energy than 19.6 J to lift that weight
because your body is not a simple machine like a lever or a pulley. To
understand why, see the faq page. In a
nutshell, the reason is that to just hold up a 2 kg mass, not move it up
at all, requires input of energy—you get tired trying to
hold up a weight at arm's length, right? And, what about lowering the
weight back down? The work done on the weight is negative which implies
that energy is being put back into you but know that it also takes
energy for you to lower the weight at a constant speed. A biological
system is considerably more complex than systems we talk about in
elementary physics classes. I think that it is of little use to try to
analyze a workout in terms of elementary physics.
QUESTION:
Speed of light again becomes 3x108 m/s when it emerges out in air from denser material without the loss of energy. Why?
ANSWER:
Just because it speeds up does not mean that it gains energy.
For light, the energy is determined by the frequency, not the speed.
When the light enters a dense medium its speed v decreases but
its frequency f stays the same. Since v=λf,
where λ is the wavelength, the wavelength decreases.
Another way to look at it is to think of the light as a swarm of
photons. The energy of a photon is hf where h is
Planck's constant, so the energy of a photon depends only on frequency.
QUESTION:
If I'm driving and hit the gas and turn left would the angle
between the velocity vector and acceleration vector be less than, greater
than, or equal to 90 degrees. I would think it's greater.
ANSWER:
You would think wrong! The velocity vector
v points
straight ahead. The tangential acceleration
at points
parallel to the velocity vector because you are speeding up. The
centripetal acceleration vector ac
points toward the center of the circle you are turning. As you can see
in the figure, the total acceleration vector
a makes an acute (less than 900)
angle with v.
QUESTION:
If the earth is curved how is it you can get a laser to hit a target at same height at sea level more then 8 km away?
How is it that it's bent around the earth?
ANSWER:
First of all, light is not bent around the earth; it travels in
a perfectly straight line and therefore, because the earth is curved,
there is a maximum distance away for a target at the same altitude. What
that distance is depends on the altitude of the laser. You say that the
laser is exactly at sea level by which I presume you mean the surface of
the earth; at this altitude you could not hit any target also at sea
level. In the figure I have drawn the earth, radius R, a point
a distance h above the earth's surface (laser location), and
another point a distance h above the earth's surface (target
location). The distance between them is 2d. Focus your
attention on one of the triangles with hypotenuse (R+h).
From the Pythagorean theorem, d=√[(R+h)2-R2]=√[2Rh+h2];
if h<<R, d≈√(2Rh). For example, if
h=10 km, about the height a commercial jet flies, 2d≈714 km is the most distant target at the same altitude which you could hit.
QUESTION:
if gravity can hold back all the seas and heavy objects to earth how can a fly or leafs move threw the air does gravity not have same force on everything I don't get how it can hold back everything but same time let small things move so easy ?
ANSWER:
For starters, the force of gravity (often called the weight) is
proportional to the mass of the object; the weight is ten times bigger
for a 10 kg object than for a 1 kg object. Second, Isaac Newton taught
us more than 300 years ago that to understand how an object moves (or
doesn't) you need to consider all forces on the object. For example, for
a leaf being blown upward, the force of the air up on the leaf is
greater than the force of gravity pulling down.
QUESTION:
I have an old fashion balance scale, center fulcrum and two dishes on either side of equal weight. If I place two weights on either side and the weights are nearly the same, the heavier side dips slightly, if the difference between the weights are large, the heavier side dips much more. I do not understand why this is so. Logic says that if there is any difference at all, the heavier side should continue to drop until it reaches a barrier to the fall no matter what the difference in the weight.
I asked a physics instructor and he did not have the answer either.
ANSWER:
The center of mass
⊗ of the scale
itself must be below the fulcrum (suspension point). Then, as shown
above, if the beam is off horizontal for the empty scale or equal
weights in each pan, there will be a restoring torque to force a balance
only for the horizontal beam.
FOLLOWUP QUESTION:
I don't think the answer given expained the phenomena described in the question. It only explained why when the weights are equal and there is a force on one side that it will bounce up again and continue to rock back and forth until it eventually evens out again. The problem I posed was a different issue, as to how the scale behaves when different weights are put on the heavier side.
ANSWER:
It does answer your question, but just not explicitly because I
did not include examples of unequal weights. So let me do that now.
Given the explanation in the original answer, I can model the scale as a
massless T with all the scale mass M located a
distance d below the pivot at the bottom of the T.
As shown in the figure, when one side is loaded with a mass m
the scale rotates to an angle θ with the
horizontal and reaches equalibrium. The sum of the torques about the
fulcrum is mgLcosθ-Mgdsinθ=0,
so θ=tan-1[mL/(Md)]. As
m gets larger, θ gets larger. For small angles, θ≈mL/(Md)
in radians.
QUESTION:
While i am reading Einstein book 'evolution of physics', i have encountered a very confusing reasoning process which i can not understand it.
The topic is about how we can deduce that the gravitational mass and inertial mass are equal just by knowing that the acceleration due to gravity for all objects at the same height is constant.
And here is the text i can not understand how he deduced that:
"Now the earth attracts
a stone with the force of gravity and knows nothing about its inertial mass. The
"calling" force of the earth depends on the gravitational mass. The "answering"
motion of the stone depends on the inertial mass. Since the "answering" motion is always the same —all bodies dropped from the same height fall in the same way—it
must be deduced that gravitational mass and inertial mass are equal.
More pedantically a physicist formulates the same conclusion: the
acceleration of a falling body increases in proportion to its
gravitational mass and decreases in proportion to its inertial mass.
Since all falling bodies have the same constant acceleration, the two
masses must be equal."
ANSWER:
Newton's universal law of gravity says that the force is
proportional to the gravitational mass of the object, F=k1mg
where k1 is a constant. Newton's second law of
motion says that the acceleration is inversely proportional to the
inertial mass and directly proportional to the applied force, F=k2mia;
in SI units, k2=1.
Therefore, a=k1mg/mi.
(This is Einstein's statement "…the
acceleration of a falling body increases in proportion to its
gravitational mass and decreases in proportion to its inertial mass.")
Now, since it is an experimental fact that a is the same regardless of
the mass, mg/mi is a constant.
If we choose to measure both mg and mi
in kilograms, and since k1 expresses the strength of
the gravitational field, mg/mi=1.
(For the field near the earth's surface, k1=MearthG/Rearth
where G is the universal gravitational constant, Mearth
is the gravitational mass of the earth, and Rearth
is the radius of the earth.)
QUESTION:
well, i saw a theory that if you rotate the ISS, which weighs 450 tons at 10 rotations per second, it would generate its own gravity. would it be possible to spin something of proportion size at a proportional speed to generate artifical gravity like the ISS?
ANSWER:
You should read earlier answers regarding "artificial gravity"
in rotating space stations. The idea is that if you are in an
accelerating frame of reference you feel like there is a force on you;
for example, when you drive fast around a curve you feel like you are
being pulled toward the outside of the curve. It is not a "theory", it
is elementary physics. The mathematics you need is that your
acceleration if you are going in a circle of radius R with
speed v your acceleration is a=v2/R.
For a given R, if you choose v such that a=g=9.8
m/s2 where g is the acceleration due to gravity, you
will feel like you are at the surface of the earth. In your case, v
can be determined from the frequency 10 rotations/second: each rotation
has a distance of 2πR so v=20πR/1. This
would imply that the acceleration would be a=9.8=(20πR)2/R=400π2R;
solving, R=.0025 m=2.5 mm. This is obviously nonsense, so you
must have remembered the the frequency incorrectly. Looking at the
figure, the main cabins appear to have a diameter of about 5 m, R≈2.5
m, so if they spun about their central axis the required speed would be
about v≈√(9.8x2.5)=5 m/s and so the
frequency would be f=5/(5π)=0.32
rotations per second=19 revolutions/minute. But this would not work at
all for the ISS because an astronaut's head would feel almost no
"gravity" because it would be very close to the axis of rotation; if she
were to lay on the floor/wall/ceiling it would be close to what it would
be like on earth. The ISS is just too small.
QUESTION:
I've been told by many that the fastest thing is vacuumed light. since light travels at different wave lengths does some light travel than other. in other words does a ultraviolet rays travel the same linear speed from A to B the same as say an x-ray. If it does than would that mean we don't have a solid speed limit of light. if they do not do we just use the longest light wave to measure the max speed, OOORRRR do we throw light like a laser... but wouldn't that still have a wave of some form.
ANSWER:
Every-day language usually interprets "light" as that which we
see with our eyes. When a physicist says "the speed of light", she means
"the speed of electromagnetic radiation". All electromagnetic radiation
travels with the same speed in a vacuum.
QUESTION:
I made a statement to somebody that a plane hitting a building was the same as if the building hit the plane at exactly the same speed,the plane now stationary. The results would be the same. In other words, if a man with large hands slapped my hand at 50 mph, it would be same as me slapping his hand at 50 mph.....its interchangable.....the other person said, no, the mass of the building and hand would have different results...
ANSWER:
Either you or your friend could be right depending on what you
mean by "different results". Let me try to set up a simple example to
demonstrate why.
-
Imagine we have a 2 lb ball of putty moving
with a speed of 5 mph striking and sticking to a 18 lb bowling ball
at rest; the time it takes to collide is 0.1 s. After the collision,
the two move together with a speed of v1. To
find v1, use momentum conservation: 2x5=(18+2)v1,
v1=0.5 mph.
-
Next, imagine we have a 18 lb bowling ball
moving with a speed of 5 mph striking and sticking to a 2 lb ball of
putty at rest; the time it takes to collide is 0.1 s. After the
collision, the two move together with a speed of v2.
To find v2, use momentum conservation:
18x5=(18+2)v2, v2=4.5 mph.
So, you see
that the two scenarios have different speeds after the colliision. But,
suppose that you were the putty ball. During the collision you feel a
force and the force is what is going to hurt you. Do you get hurt as
badly, not as badly, or equally as badly during the collision? What
determines the force you feel is the acceleration you experience during
the collision, how quickly your velocity changes, which is your final
velocity minus your initial velocity divided by the time of the
collision.
-
For the
putty ball moving initially, (vfinal-vinitial)/t=(0.5-5)/0.1=-45
mph/s.
-
For the
bowling ball moving initially, (vfinal-vinitial)/t=(0-4.5)/0.1=-45
mph/s.
You could go
through through the exact same process to find that the bowling ball
experienced exactly the same force regardless of who moved initially. A
physicist would say that you were right, but the ambiguity of your
statement means that the other guy could split hairs. As far as physics
is concerned, the only thing which matters is the relative
velocities of the two before the collision. If the putty ball were
moving 105 mph and the bowling ball were moving 100 mph in the same
direction, the result of the collision which matters (the force) would
be the same.
QUESTION:
The atmosphere is heavy. If the weight of a column of air above your desk is about the same weight as the bus you rode to school in, why doesn’t air pressure crush your desk?
ANSWER:
Because the atmospheric pressure also acts up under your desk.
QUESTION:
I have always wondered how much energy do you do with if you let a kettle at 1800 W be running for two minutes? What is the approximate cost for this?
this is not a homework question.. just a question i wonder =)
ANSWER:
A
Watt is one Joule per second, 1 W=1 J/s. Energy consumed by an 1800 W
kettle in 2 min is 1800x120=216,000 J. But, we are more used to
measuring electrical energy in kilowatt hours, (1 kW·hr)(1000
W/1 kW)(3600 s/1 hr)=360,000 J. So the energy used by the kettle is
(216,000 J)/(360,000 J/kW·hr)=0.6 kW·hr. A kW·hr
costs on the order of 5¢-15¢, so the cost would be between 3¢
and 9¢.
QUESTION:
Why do unstable elements always give off alpha particles with 2 protons and 2 neutrons. Essentially a Helium nuclei. Why not a hydrogen nuclei or a heavier nucleus
ANSWER:
Alpha-decay is only prevalent in very heavy unstable elements. Most
unstable nuclei decay by beta decay, the ejection of an electron or
positron along with a neutrino. The reason that alpha-decay happens is
that the alpha particle is an extremely tightly bound particle and
therefore there is a fairly high probability that it will spontaneously
form inside a very heavy nucleus where there are a lot of neutrons and
protons to contribute. For a more extensive discussion, see an
earlier answer.
QUESTION:
Every day my wife reads the latest fake news about planet 9 and sits weeping in fear the whole time . As a scientist ,surely you can tell me one thing that just shuts this whole fiasco down . Please. Tell me that undeniable fact that will convince her that we are in fact safe from a collision or near miss from this nonexistent space oddity and it 's cohorts . It s affecting her health . And I love her . So it' i m feeling her pain as well .
ANSWER:
First of all, "Planet 9" is a serious astronomical topic. Minor
irregularities in the orbits of some of the distant planets suggest the
presence of another planet farther out.
Serious efforts are underway to try to observe it. But its
anticipated period is more than 10,000 years, so if it is far away now,
it is not likely to be a problem for us for far longer than the lives of
any of us. All reputable astronomers have declared that (if it actually
exists) it is absolutely no danger to earth. Google "planet 9" to get
lots of good information. But, even if there were a planet much closer,
like the "fake news" "planet x", there is an amazingly low likelihood of
its colliding with earth. Most lay folk like your wife have no
comprehension of the vastness of space; the probability of two
particular objects in the solar system colliding is, for all practical
purposes, zero. It is often posited that a "rogue planet" passing
through the asteroid belt would "shake loose" a "storm" of asteroids
toward the earth. I recently
answered a question
addressing this possibility which you might find useful.
QUESTION:
Can tension ever be negative?
ANSWER:
This
is an ambiguous question. If you mean can the magnitude of the tension
force exerted by a string be negative, the answer is no; a string can
only pull, it can never push. But, if you mean can the tension ever have
a component which is negative, the answer is yes; it simply depends on
how you have chosen your coordinate system. But, if you have drawn the
tension vector T such that the string is
pulling on something and you solve the problem and the magnitude T<0, you have made
a mistake somewher.
Q&A OF THE
WEEK (1/21-28/2017)
QUESTION:
My question is about the maximum tension experienced by a bow string. I'm
specifically concerned with a traditional or recurve bow NOT a compound bow
with pullies. I want to know the max tension compared to the draw weight so
I have an idea how strong to make my strings. So here's the scenario, what's
the maximum tension in the string for a recurve with a 70 lbs draw weight
and a physical weight of 25 oz? I'm assuming the maximum tension is when
it's at brace (not full draw), right after the arrow leaves. I think this
because not only does the string have to oppose the restoring force of the
bow limbs but it also has to stop the momentum of the limbs that isn't
transferred to the arrow.
ANSWER:
To compute the tension I would need to know the geometry of the bow. I can
tell you that the tension will be at the maximum draw for a simple bow, not
where the arrow leaves the string.
REPLY:
The bow is 64 inches long and the string is about 4-5 inches shorter than the bow. It Is braced at 6 inches and has tips that are 3 inches recurved behind the handle. Its draw length is 28 inches and has an elipitcal/circular tiller shape at full draw.
ANSWER:
(An incorrect answer was posted earlier. This is a reposting, correct
now, I hope!)
When researching the physics of archery I discovered that this can be a
very complicated problem requiring very sophisticated numerical
calculations on computers if you want precision descriptions of all the
details. You, however, require only a rough calculation for estimating
the strength of the string. I can do that and it is more appropriate for
the spirit of this site—to solve problems with simple physics
concepts. This problem requires facility with trigonometry,
understanding of Hooke's law, and application of Newton's first law. The
simple model I will use was one used before the advent of computers; the
bow is modeled as two straight rods (purple) the ends of which move on a
circle as the string (red) is drawn. With this simple model, most of the
details of your bow are not necessary. When the string is braced
(undrawn) there is a certain tension in the string and this tension will
increase as the bow is drawn. So, the maximum will be at the maximum
draw. The figure shows, roughly to scale, the situation. Using simple
trigonometry (law of cosines), I find β=59.60.
The point where the draw weight W is being applied must be in
equilibrium, W-2Tcosβ=0; solving, T=69.2
lb. I think that your concern about the string having to "stop the
momentum of the limbs" is misplaced because bows tend to be quite
elastic so that nearly all the energy imparted to the bow by drawing it
is imparted to the arrow and the limbs will end nearly at rest. Not so
if you draw and release without an arrow, though; what I have read is
that in that situation you are more likely to break your bow than the
string. I am working on a general solution which I will later add here
but thought I would post the part of the solution which answers your
question regarding the tension in the string.
GENERALIZED
SOLUTION:
To get a better understanding of this problem it is worthwhile to
find an analytical solution for the tension as a function of draw
distance. My research showed me that a traditional or recurve bow
behaves, to an excellent approximation, like a simple spring
(Hooke's law), the draw weight being proportional to the draw distance,
i.e. W≈kx where x is the distance the
string is drawn and k is the spring constant. In this case, since W=70 lb when x=28
in, k=2.5 lb/in. Using the law of cosines, cosβ=(L2+(x+d)2-R2)/(2L(x+d)).
Again, the point where the force W is applied
is in equilibrium so W-2Tcosβ=0 or T(x)=kx/(2cosβ).
Now, note that in the limit where x → 0, β → 900
and cosβ → x/L. Therefore T(0)=kL/2.
Using your numbers, T(0)=37.5 lb and the angle and tension for
all points are plotted below. At full draw, β=560
and T=64 lb. It was interesting to me that in order for the
calculated values to be correct at zero draw, very precise relative values of
R and L
had to be used because otherwise the expression for cosβ would not be 0 exactly when x=0. The value was
R=30.59411708 inches for L=30.
QUESTION:
If a charged particle oscillates, it produces a propagating electromagentic wave. What happens when the motion of the charged particle is not oscillatory, but Brownian? Is the emitted radiation much weaker?
ANSWER:
It is acceleration which causes radiation. An oscillating charge is always accelerating except at the instants it passes through its equilibrium position so continuously radiating. Brownian motion is a series of very brief accelerations followed by much longer periods of constant velocity and therefore
the radiation is a series of pulses. There is no way you can predict
which is weaker by only knowing the acceleration. It depends on the
magnitude of the charges involved, the magnitudes of the accelerations,
and the frequencies of the accelerations.
QUESTION:
My question is related to photons. We have coherent light IE laser emission which over distance spread out much more slowly. Our sun emits incoherent light, which is the same for all observant stars. My query is why do we still see the stars as pinpoints of light no matter if they are near or very distant.
ANSWER:
The reason all the stars look like pinpoints is that they are very far
away, not because they have tightly collimated beams like a laser; if
the sun were very far away it would look that way too. If the light you
see from a star were a tightly collimated beam, it would be pointed
directly at you and if you stepped to the side it would disappear; you
would see almost no stars at night because nearly all of them would be
pointing elsewhere.
QUESTION:
when i kick the ball ,i exert a force on it and it exert the same force on me on the opposite direction according to 3rd law of newton ,bet what is the effect of the ball on me .the effect of the force that i exert on the ball make it move ,but my leg do not move .
ANSWER:
Do you feel it when you kick the ball? Of course you do and what you are
feeling is the force the ball exerts on your foot. But your foot moves in
response to all the forces on it, not just that one. During the kick
your leg is exerting a force on your foot much larger than the ball's
force on you. Your foot will be moving a little bit more slowly after
the kick than if you had not kicked the ball.
QUESTION:
With the understanding that "gravity" is a fictitious force created or experienced by Earths acceleration what I don't understand is what is the source of this acceleration? How is it that the Earth is continuously accelerating us upwards to produce weight?
ANSWER:
Your understanding is wrong. Gravity is not a fictitious force caused by
acceleration. Maybe it would help you to read some of my earlier answers
regarding gravity and general relativity listed on the
faq page.
QUESTION:
This is odd, but My family has just moved into a huge house with little outdoor space. We live in a climate that is cold in the winter, and I want my children to get some exercise on a daily basis. We own a trampoline, and have space for it indoors on the Second floor of our house. The ceilings are 12 feet high, so there would be no problem with the kids hitting their heads on the ceiling. My question is whether or not the house would stand up to the force generated by the trampoline. The walls of the house are made of concrete (you can't nail into it.) I am assuming the floors are quite solid as well, as they must support the weight of the house. They are concrete as well.
My Youngest child is quite large (6 ft, 260 lbs)--he is only twelve. We need the activity.
ANSWER:
First, a disclaimer: I can give you an idea of how much force the floor
will experience. I cannot predict whether this will cause your floor to
fail because I have no information about your floor other than that it
might be concrete. I have watched some videos and it seems that the
jumper never goes as high as h=2 m and the trampoline never
goes down as far as s=1 m. So I will just do my calculations
with those to get an upper limit on what force might be expected. Your
son's mass is about m=120 kg. An object falling from h=2
m will hit the trampoline with a speed of about v=√(2gh)≈√(2x10x2)=6.3
m/s. I will treat the trampoline as a simple spring so that I can write ½mv2=½ks2-mgs
where k is the spring constant. Putting in m,
v, and s and solving for k I find k=7200
N/m; since the force exerted by a spring is F=ks,
the largest force the trampoline exerts on your son is about 7200 N=1600 lb;
Newton's third law tells you that this is also the force your son exerts
down on the trampoline. Therefore, the trampoline exerts a force down on
the floor of 1600+W where W is the weight of the
trampoline. This is a little more than the weight of a grand piano.Keep
in mind that this is the greatest force and just for an instant; the
average force over the collision time would be half this. This is a
little more than the weight of a grand piano.
QUESTION:
I recently watched with great interest a PBS program which described in layman's terms how Uranium 238 transforms into the different chained elements, to include U235. It also explained the basics of the chain reaction caused by splitting U235 using E=MC^2 as the basis for energy release. This is where I was a little unclear.
The split was described as one U325 nucleus splitting into two separate nuclei with some individual particles released (can't remember if they are protons or neutrons) Those particles then collide with other U235 atoms in proximity triggering subsequent splits and particle releases as part of a chain reaction.
My question is that the mass doesn't appear to be transforming into energy (E=MC^2). Rather it appears that it is simply splitting and being cast off, so what causes the energy release? This assumes that the number of particles in the remaining two nuclei + the particles independently released still equal 235. There was a general reference in the program to how the Strong Force reduces the size of the resultant smaller nuclei, but it didn't say if matter within each was converted to energy or if the number of particles are additionally reduced through such a conversion. Thanks for any clarification you can provide.
ANSWER:
Suppose that you weigh one 235U and one neutron. Now, when
you add the neutron to the 235U it fissions and, after all is said
and done you have two lighter atoms and a few neutrons; if you weigh all
these byproducts, you will find that approximately 0.1% of the original
weight is missing. Where did it go? Most of it went into kinetic energy
of the byproducts, that is they are all moving faster. Kinetic energy of
atoms is essentially what thermal energy is—the reactor (or
bomb) has gotten hotter. For a bomb it all gets enormously hotter
resulting in the explosion. For a reactor, the rate of fissioning is
controlled and the heat is extracted to drive turbines to create
electricity. More detail can be found in an
earlier answer.
FOLLOWUP QUESTION:
I'm assuming (correct me if I'm wrong) that such motion was some percent of the speed of light which would account for the transformation of about
0.1% of its matter to energy.
ANSWER:
First things first—there was an error in my original answer, now
corrected throughout: the amount of mass converted to energy is
about 0.1%; nuclear fusion is about 1%. No, the motion of the atoms is
nowhere near the speed of light. It is simply classical ½mv2
type of kinetic energy. The "transformation" is simply that—mass
energy transformed into kinetic energy. To understand, see a
recent answer which explains why a bound system has
less mass than if it is pulled apart and the mass measured.
It turns out
that heavy nuclei like uranium are less tightly bound than nuclei with
roughly half their mass; therefore when they split the products are less
massive. That is why fission works as an energy source.
QUESTION:
If a capacitor is made of oppositely charged plates, why do they look like cylinders inside computers, remote control cars and other electronics
ANSWER:
The easiest form of capacitor to understand and analyze in an
introductory physics class is the parallel plate capacitor. But any two
conductors insulated from each other is a capacitor. One possible
capacitor is a wire along the axis of a hollow cylinder, but that is not
what the common capacitor you are referring to is. Rather, it is a parallel
plate capacitor! The analysis of the capacitance of two parallel plates
shows that the capacitance is proportional to the area of the plates and
inversely proportional to the distance between them. So, take a two
ribbons of foil as long as a football field to make the area big and
separate them by sandwiching a ribbon of mylar between them to make the
separation small; then just roll it up so you can fit it in your device!
QUESTION:
Suppose a block is moving with constant velocity towards right on a frictionless surface and during its motion another block of slightly smaller mass lands on top of it from a negligible height.
I argue that the lower block will eventually start moving to the left and upper block will end up moving towards right provided that there is friction between the blocks but not between lower block and ground . My friends can't accept my reasoning. Am I wrong? Please help!
ANSWER:
I hate to tell you, but you are wrong. This is actually a simple
momentum conservation problem. Call the masses of the upper and lower
blocks m and M, respectively. Before they come
together the momentum is Mv where v is the incoming
speed of M. When the masses come in contact they will slide on
each other but, because there is friction, they will eventually stop
sliding and both will move with a velocity u; the linear momentum will
now be (M+m)u. Conserving momentum, u=[M/(M+m)]v.
They both end up going with speed u and move to the right.
You can also determine the time it takes for the sliding to stop. m
will feel a frictional force to the right of magnitude f=μmg
and M
will feel a frictional force to the left of magnitude f=μmg
(Newton's third law). So, choosing +x to the
right, the acceleration of m is a=μg and
the acceleration of M is A=-μg(m/M).
The velocities as a function of t are vm=μgt
and vM=v-μgt(m/M);
we are interested in the time when vm=vM,
so solving for t, t=Mv/[μg(M+m)].
If you substitute this back into vm or vM,
you will find the same value we found for u above: vm=vM=u=[M/(M+m)]v.
FOLLOWUP QUESTION:
If the block of mass M is not too long, i.e., the total distance that the upper block can slide is less that the distance it could move in your calculated time " t" , wouldn't the two blocks get separated?
ANSWER:
Well, of course the block has to be big enough, otherwise m
will drop down on the frictionless surface. It would be a good exercize
for a student to calculate how far the block would slide for some
μ. And then, if this is greater than the size of the bottom
block, how fast will each be moving after separating.
QUESTION:
What would it take for a falling body to travel 25' horizontally from a 350' height?
I'm sorry, I didn't explain well. This is an actual situation. There is a guard booth at the base of the bridge, 25' away. If someone was to jump off the bridge at a height of approx 300+ feet, would it be possible that they would be able to strike the booth?
ANSWER:
One possibilty is if there is a steady wind blowing in the right
direction. You should know that calculation of air drag, the force of
the wind in this case, is always a rough estimate, not something you can
predict with precision. I prefer to work in metric units, but I will
switch back to ft and mph at the end. A rough estimate of the wind force
is F≈¼Aw2 where A
is the area presented to the wind and w is the speed of the
wind (this approximation only works for SI units). I will assume that
the horizontal speed acquired by the jumper is small compared to the
wind speed so that w is a constant during the fall. So,
approximating A≈1 m2, the
acceleration horizontally is ax=F/m=w2/(4m)
where m is the mass of the jumper. The time to fall 350 ft is
about t=4.7 s and the horizontal distance is then x=½axt2=w2t2/(8m).
Now, x=25 ft=7.6 m and I will take m=150 lb=68 kg.
Solving for w I find w=13.7 mps=31 mph. A steady wind
of 31 mph could cause the jumper to move 25 ft horizontally.
You have not told me where the the booth is. If it is under the bridge,
then the jumper could not propel himself in that direction. But if the
booth is 25 ft out from under the bridge, the jumper could jump out as
well as drop down. The speed vx he would have to
give himself horizontally can be easily calculated: vx=x/t=25/4.7=5.3
ft/s=3.6 mph.
QUESTION:
In electromagnetism we compute the intensity of a wave by taking the square of its amplitude. Why do we not do exactly the same thing with quantum mechanical waves?
ANSWER:
Actually, you could say that is exactly what we do. You just have to ask
what intensity means for the wave function. In electromagnetism,
intensity is just the energy density flux, the power per unit area,
measured in W/m2. The square of the wave function is the
probability density and so is a measure of the likelihood of finding the
particle in one small volume in space. If you add up the square of the
wave function at all points in space (integrate), you must get the
answer of 1 because the probability of finding the particle somewhere in
space must be 1 for this interpretation to make sense; this is called
normalization.
QUESTION:
My dad told me about your website, very interesting reading. My question deals with molecules. When a molecule emits a photon, the mass of the molecule decreases to account for the energy in the photon. So, the mass of the molecule as a whole decreases, but this mass does not come from the "parts" of the molecule. In other words, the mass of the constituent electrons does not decrease, the mass of the protons does not decrease, so the energy must come from the electric field between the electrons and protons.
But the electric field has energy, not mass. Now mass is a form of energy, but I don't think you can say that the field has mass? But yet, it is said that the mass of the molecule decreases. The electric field contributes to the mass of the molecule, but yet it is incorrect to say that the field has mass?
ANSWER:
Actually, this is not as complicated as you are trying to make it. It
all boils down to the fact that mass is a form of energy and must be factored into
any energy conservation that occurs in an isolated system. You say that
the masses of the protons and electrons do not change, but that is not
right. Look at the simplest case, a hydrogen atom. If you measure the
mass of this atom it will be less than if you measure the masses of a
free electron and a free proton. Here is how you can see that: if you
pull the electron away from the proton, that is you ionize the atom, do
you have to do any work? Of course you do because the electron and
proton are bound together. So, you have added energy to the system (p+e)
and that energy shows up as mass. In a system as complicated as a
molecule you cannot say which particle or particles changed their
masses, but you can say for sure that the total mass of the molecule
changed by exactly the energy of the emitted photon divided by c2.
QUESTION:
This is my 2nd question related to the issue of time dilation - this one being related to the issue of motion [the other being based on gravity]. Since time dilation occurs for all moving objects, and considering further that the Earth has been revolving around the sun at 30 km per second for the last 4 billion years -- and further that our solar system is moving at roughly 45K mph through space, can't it be said that, compared to other objects in the universe, time dilation has occurred to a significant degree for our planet over those 4 billions years? And that really every object in the universe likewise has its own unique time dilation associated with it? Can't it also be said that every consolidated arrangement of matter in the universe is moving along at different "rates of time?" Wouldn't, over the course of several billion years, these "pockets" of different time spans become more and more "incompatible" with each other?
ANSWER:
The two speeds you quote are about the same (45,000 mph≈2x104
m/s and 30x103 km/s=3x104 m/s). So let's just
choose the larger one and see how much time dilation there is. Relative
to the sun, an elapsed time T=4x109 y would
correspond to T'=ΥT wwhere Υ=1/√(1-(v/c)2)=1/√(1-(3x104/3x108)2)≈(1+0.5x10-8).
Therefore T'≈(T+20), a difference of 20 years. This
may sound like a pretty long time to you, but relative to 4 billion it
is less than 10-6 %. That is not "significant" to my mind.
You are right that every object has its own clock which, relative to
other clocks, is not necessarily the same; every object also has its own
meter stick, not necessarily the same as other meter sticks in the
universe. The important thing is that you always must talk about
velocity with respect to what.
QUESTION:
The movie "Interstellar" did a nice job of explaining how time dilation works in a massive gravity field. My question relates to how we on Earth measure the age of the universe to be 14.3B years. If I could make that measurement on the planet orbiting the "Interstellar" singularity [and since, theoretically, if I could view life on the singularity planet from Earth, it would all look to be in super slow motion], what would my measurement of the universe's age be from the my perspective on that planet? If time moves more slowly on the singularity planet, wouldn't my estimate of the universe's age be much less?
ANSWER:
As I say on the site, I do not usually answer questions on
astronomy/astrophysics/cosmology. Maybe I can make a stab at this. The
microwave background radiation which pervades the universe is generally
considered the best source for information about the big bang and
measurements are probably the best determinations of the age of the
universe. In your high-gravity position, you would see the same
microwave radiation I do. Likely, you would have to make corrections to
your observations due to the intense gravity, but you would still
conclude the same age as I did.
QUESTION:
hi, can and are earthquakes be caused by celestial alignments ie planets?
ANSWER:
Let's take a simple example. As seen from earth, Mars and Jupiter are
aligned. I estimated the force on a 1 kg object which is sitting, let's
say, on the San Andreas fault: F=3x10-11 N; the
weight of that 1 kg object is about 10 N. I would say that putting a 1
kg object on the ground is a great deal more likely to cause an
earthquake than those planets, wouldn't you?
QUESTION:
So there's a powerline outside my bedroom window, and I thought, huh. Turns out I'm sleeping with my head in a 6mG AC magnetic field (according to two meters). Help me use physics to stop caring.
How do I estimate/calculate which puts more force on the charged particles (calcium, potassium, sodium) in my brain: a) An aqueous solution at 98.6 degrees Fahrenheit or b) a magnetic field acting on charged particles moving at some estimated speed in said aqueous solution.
My hope here is that the force of (b) is like an order of magnitude or two, or something, below the "noise floor" of (a) and then I can stop caring forever.
ANSWER:
How about this: the earth's magnetic field is about 0.6 G, two orders of
magnitude bigger than the field due to the power line, and you are
exposed to it 24 hours a day. It is also possible that there is some
other source of field closer by than the power line which, though a much
smaller current, would produce a much bigger field. For example, if
there were a wire in the wall carrying a typical household current of 1
A, the field 2 m away would be 1 mG. There is no good scientific
evidence that any magnetic fields you are likely to encounter have any
effect, good or bad, on the functioning of your body.
QUESTION:
I do not want a theoretical answer, but has any experimentalist ever put a very sensitive weight balance below a vacuum chamber before and after vacuating it? Does it get lighter or... heavier? I do not have a sensitive balance nor a vacuum chamber.
The reason I ask is that it would say something about the density, or absence of density, of the vacuum.
If I understand pressure correctly, the scale would read a smaller weight value, due to less gas being in the column of air directly above it, but there might also be new physics there, if it is not the case. I simply do not know.
ANSWER:
You do not want a "theoretical answer" but you clearly do not understand
the physics so I am obliged to give you one anyway. Let us assume the
simplest possible "weight balance" so that I do not have to worry that
it might operate differently in a vacuum. Envision just a simple string
with a tiny butcher's scale which will measure the tension in the string
and then hang an unknown weight of mass M and volume V
from the string. Besides the string, there are two forces on the object
being weighed, its weight Mg and the buoyant force B=ρVg
where ρ is the density of air (about 1 kg/m3 at
atmospheric pressure) and g=9.8 m/s2 is the
acceleration due to gravity. The scale will read W=Mg-B, an
incorrect measure of the weight. Putting the whole device in a vacuum
will change B to zero because the air is gone, so W=Mg,
the correct weight. To get an idea of how important this is, consider
weighing a solid block of iron whose mass is 1 kg. The density of iron
is ρiron=7870 kg=M/V, so the
volume of the block is V=1/7870=1.27x10-4 m3.
So, the true weight is W=9.8 N and the measured weight in air
is (9.8-1.27x10-4) N=9.7999 N, an error of 0.0013%. However,
there are certainly examples where the effect of buoyancy would be very
important. For example, consider an air-filled balloon. I did a rough
calculation and estimated that the volume of an inflated balloon is
about 5x10-3 m3 so it contains about 5x10-3
kg of air; the mass of an uninflated balloon is about 5 gm=5x10-3
kg, so the total weight of the inflated balloon is about 9.8x10-2
N. But if you weighed it in air you would only measure half that amount.
Your question, has anybody ever actually observed this, is a no brainer:
since the existence of a buoyant force has been known and understood for
well over 2000 years (Archimedes' principle), anyone
wanting to make an
extremely accurate measurement of a mass would either correct for it or
eliminate it.
QUESTION:
Several years ago, I was caught in a massive windstorm in a skyscraper. I was on the 54th floor (approx. 756 feet from street level, full building height is 909 feet) , pulling cable, and I stopped for a break. I left a cable pulling string hanging from the ceiling (48 inches free hanging length) in the office, with a 1/4 lb weight attached, and when the storm hit, the weight began swinging like a pendulum. The arc was 16 inches (eyeballing it), and traversing the length of the arc took about 1 second. How can I calculate how far (full arc) the skyscraper was moving by observing what the pendulum in the building was doing?
ANSWER:
A 48" pendulum has a period of about 2.2 s, the time to swing over the
arc and back. Since you were estimating, the pendulum was swinging with
about the period it would if the building were not moving at all. I
would conclude that either the pendulum got swinging somehow and the
building was not perceptibly moving or that the period of the building's
motion was about the same. If the building was swinging with a period
significantly different from 2 s, the pendulum would be swinging with
that same period; that is called a driven oscillator.
