Is uniform circular motion perpetual?

Main Question or Discussion Point

If i took my pencil to a place in space without gravity or air resistance and i spun it, would it spin forever? I mean its undergoing a centripetal acceleration, so energy has to come from somewhere to keep it spinning right? (and of course this is not meant to be an idea for perpetual motion, i know those are against the rules)

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SteamKing
Staff Emeritus
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Things move for a long time in outer space not because they are always consuming energy, but because the vacuum of space offers no resistance to movement, unlike on earth. If you spin a pencil in space, the energy imparted in the initial spin is not diminished by any friction acting on the pencil due to its spinning motion. As Newton said, "A body in motion tends to stay in motion, unless acted upon by an external force."

Simon Bridge
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If i took my pencil to a place in space without gravity or air resistance and i spun it, would it spin forever?
If nothing else happens - yep.

I mean its undergoing a centripetal acceleration, so energy has to come from somewhere to keep it spinning right?
Nope. The input of energy occurred at the start when you spun it up.

(and of course this is not meant to be an idea for perpetual motion, i know those are against the rules)
This sort of perpetual motion is not against the rules ... it's one of Newton's Laws.

The prohibition is not against perpetual motion, it is against perpetual motion machines - the idea that you can get useful work out of something like this. The maximum amount of energy you can extract from the spinning pencil is the amount of work you did making it spin.

Donald Simanek has an excellent resource on these things.
http://www.lhup.edu/~dsimanek/museum/physgal.htm
... see the bit about rotations and wheels.

Hmm. I always thought a constant force was necessary to keep something constantly in circular motion even in absence of friction...

WannabeNewton
That's when you have something like a ball tied to a string and need a means of keeping the ball afloat in the air in a circular trajectory by applying a sufficient tension across the string while you stand on the surface of the Earth; this is a case of a controlled circular trajectory about some central point. The OP is instead talking about a situation where you given an initial spin to a pencil in deep isolated space; there are no subsequent external forces at work on the pencil hence nothing to disturb that initial spin.

A.T.
Hmm. I always thought a constant force was necessary to keep something constantly in circular motion even in absence of friction...
Force is not energy. The centripetal force in uniform circular motion is perpendicular to velocity, so it doesn't do any work on the object.

Force is not energy. The centripetal force in uniform circular motion is perpendicular to velocity, so it doesn't do any work on the object.
I was just saying that there would be no force acting on it after it is released, yet it continues spinning..

davenn
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I was just saying that there would be no force acting on it after it is released, yet it continues spinning..
yes.
particularly the last sentence :)

Dave

CWatters
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Hmm. I always thought a constant force was necessary to keep something constantly in circular motion even in absence of friction...
Perhaps it helps to think of circular motion as having two components, one tangential and the other radial.

A force is required to produce the radial component but since the radius is constant that force does no work.

No force is required to maintain the tangential component (there is no air resistance in space).

Nugatory
Mentor
I was just saying that there would be no force acting on it after it is released, yet it continues spinning..
There's no net force acting on the entire pencil, so its center of mass isn't moving; the pencil is spinning in place.

There are forces acting on the ends of the pencil. It's a solid object so it resists stretching, bending, changing shape. Without these forces the tip and eraser end of the pencil would go moving off in different directions; with these centripetal forces they're pulled into circular motion around the center of the pencil.

However, these forces aren't doing any work because they're acting along the length of the pencil and the pencil is rigid so doesn't change its length.

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jtbell
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Hmm. I always thought a constant force was necessary to keep something constantly in circular motion even in absence of friction...
The forces between the molecules in the pencil keep those molecules moving in a circular path instead of flying off in a straight line.

That's when you have something like a ball tied to a string and need a means of keeping the ball afloat in the air in a circular trajectory
A ball tied to a string attached to a fixed point, and whirling around in a circle, eventually "spins down" because of air resistance (in the earth's atmosphere) and energy losses in the flexing of the string at the attachment point. If you do this on the moon, you eliminate the air resistance, but the flexional losses remain. If you could come up with a way of supporting the string at the center in a way that does not involve any friction or flexing of the string, and do the whole thing in a vacuum, the ball would whirl around forever just like the pencil in space would spin forever.

D H
Staff Emeritus
However, these forces aren't doing any work because they're acting along the length of the pencil and the pencil is rigid so doesn't change it's length.
The pencil would continue rotating even if it wasn't rigid (which it isn't; there is no such thing as a truly rigid body).

The forces that hold the pencil together don't do any work on the pencil because they are internal rather than external forces. Internal forces that are subject to Newton's third law hold the pencil together but they can't change the pencil's total linear or angular momentum.

Hmm. I always thought a constant force was necessary to keep something constantly in circular motion even in absence of friction...
It's those internal forces that hold the pencil together that keep the pencil rotating.

What's been missing from the discussion so far is the concept of angular momentum. Angular momentum is a conserved quantity. An external torque must be applied to the pencil to change its angular momentum. You've ruled out that external torque in the original post, so in a Newtonian world, the pencil must keep on rotating forever to keep that angular momentum constant.

ehild
Homework Helper
If i took my pencil to a place in space without gravity or air resistance and i spun it, would it spin forever? I mean its undergoing a centripetal acceleration, so energy has to come from somewhere to keep it spinning right? (and of course this is not meant to be an idea for perpetual motion, i know those are against the rules)
The spinning pencil does not perform circular motion: but all it points do. A small piece of pencil at distance r from the CM is held at that distance by internal forces acting among the molecules. These internal forces provide the centripetal force to the circular motion of that piece.
The whole pencil just rotates about its CM, with the angular speed you gave it initially. It will keep its angular momentum forever if no external torque acts on it.
You can make the pencil perform circular motion in the space by bringing it close to a massive body and give it the appropriate initial velocity for a circular orbit. Gravity would provide the centripetal force .

ehild

This discussion has been very informative. I had never before thought about the dynamics of circular motion where the center of mass is stationary.

I thank each and everyone of you! :)

A.T.
There's no net force acting on the entire pencil, so its center of mass isn't moving
There's no net force acting on the entire pencil, so its center of mass isn't accelerating. Mere movement doesn't require a net force.

sophiecentaur
Gold Member
Whatever forces are acting, no energy is 'consumed' or transferred if there is no movement in the direction of those forces (i.e. the string / pencil gets no longer or shorter). Work done is Force times Distance moved in the direction of the force.

Nugatory
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There's no net force acting on the entire pencil, so its center of mass isn't accelerating. Mere movement doesn't require a net force.
Yes, of course you're right. This happens sometimes when I post before the second cup of coffee... Jd0g33, I hope you read what I meant to say ("accelerating") and not what I did say ("moving").

jfizzix
Gold Member
If i took my pencil to a place in space without gravity or air resistance and i spun it, would it spin forever? I mean its undergoing a centripetal acceleration, so energy has to come from somewhere to keep it spinning right? (and of course this is not meant to be an idea for perpetual motion, i know those are against the rules)
It would indeed spin forever, and according to newtonian physics it would keep spinning at the same rate (as is seen with the conservation of angular momentum).

Things get a little weird when we bring relativity into the picture.

What happens is that the pencil is spinning (that is, since the velocity of the atoms are not all the same and keep changing in time), the pencil loses energy as gravity waves.

The rate that energy is being lost depends on how fast the pencil is spinning, so as it slows down, it is also losing energy more slowly. As time stretches on, the pencil never truly stops, but it slows down nearly to zero spin. This doesn't violate any conservation laws, since this angular momentum is transferred to the gravity field.

The same thing happens in classical electromagnetism, where charged electrons orbiting an oppositely charged nucleus lose energy as electromagnetic waves, and the electrons spiral into the nucleus in about a hundreth of a nanosecond. Of course, that's the classical theory. Quantum mechanicsally, this doesn't happen because the nature of an electron is (apparantly) not to have a well defined position and momentum. There is a minimum energy the electron can have orbiting a nucleus, and this is known as the ground state.

Compared to the electromagnetic force, gravity is extremely weak, so it would take eons for the pencil to slow down appreciably.

What happens is that the pencil is spinning (that is, since the velocity of the atoms are not all the same and keep changing in time), the pencil loses energy as gravity waves.

The rate that energy is being lost depends on how fast the pencil is spinning, so as it slows down, it is also losing energy more slowly. As time stretches on, the pencil never truly stops, but it slows down nearly to zero spin. This doesn't violate any conservation laws, since this angular momentum is transferred to the gravity field.
So you are saying that a rotating object loses energy/momentum through "gravity waves", and a non-rotating object does not?

I thought that rotational motion is just a special case of linear motion. The difference being that a body in uniform circular motion is undergoing acceleration. So is it the acceleration of rotational motion that is the cause of the energy loss as gravitation waves?

That would indicate to me that there would be a similar energy loss of a linearly accelerating body due to the gravitational waves that you mention. Is this so?

mfb
Mentor
So you are saying that a rotating object loses energy/momentum through "gravity waves", and a non-rotating object does not?
That is right.

I thought that rotational motion is just a special case of linear motion. The difference being that a body in uniform circular motion is undergoing acceleration. So is it the acceleration of rotational motion that is the cause of the energy loss as gravitation waves?
Right.
More specific, it is the variable quadrupole moment. A perfectly uniform ring could rotate without emitting gravitational waves, for example.

That would indicate to me that there would be a similar energy loss of a linearly accelerating body due to the gravitational waves that you mention. Is this so?
Right.

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Thanks mfb, and quick reply too.

phinds
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If i took my pencil to a place in space without gravity ...
Just to be sure you are clear on this, there IS no such place. There are places where gravitational attraction is miniscule, but there is no place where it is zero.

Simon Bridge
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@jd0g33: was any of that any use?

D H
Staff Emeritus
It would indeed spin forever, and according to newtonian physics it would keep spinning at the same rate (as is seen with the conservation of angular momentum).
No, it wouldn't.

First off, even if angular momentum was conserved (which it isn't; more below), the pencil would not keep spinning at the same rate. It's angular momentum, not angular velocity, that is conserved sans any angular momentum transfer.

A pencil is ideally a rigid body with one axis of symmetry. A rigid body that is not subject to external torques will only continue to rotate about the same rate, forever, if it is the axis of rotation is perfectly aligned with one of the body's principal axes. An object such as our ideal pencil has one principal axis along that axis of symmetry and an infinite number of principal axes normal to the symmetry axis. Getting the rotation perfectly aligned with one of those principal axes is an impossible task; it's a space of measure zero. An ideal pencil would tumble but with a constant angular momentum.

In reality, a pencil is a non-rigid body with no axes of symmetry. The rotation builds up internal stresses, and it's not perfect. The pencil heats up a bit and radiates this heat away. The pencil loses energy. This means the pencil's rotation will migrate to being about the axis with the largest moment of inertia.

The radiation almost certainly won't be spherically distributed. The asymmetry means that the radiation will transport angular momentum as well as energy from the pencil. The pencil's rotation rate will slowly slow down -- but not near as slowly as the puny gravitational radiation that results from general relativity.

mfb
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@D H: I would expect that this process slows down the rotation initially, but it stops as soon as the axis of rotation is the largest principal axis of the pencil. Afterwards, it does not influence the rotation speed any more.