Tetherball radius doubling: why does momentum seem non-conserved?

[eosphorus]

i have a tetherball that goes from a radius of 1 m to a radius of 2 m.

the initial speed is 1 m/s, what's the final speed?

if its 0.5 m/s the energy of the ball halfs so it must be transferred to Earth via pole thickness so the Earth gains kinetic energy so gains momentum,so the momentum of the whole system grows

if its 1 m/s then conservation of momentum fails because momentum has doubled and besides some torque has been applied to Earth via the pole thickness

 

[krab]

It's the second case; the ball's speed does not change.

It is difficult to get a feel for this situation because the tether pole's radius is so very small compared with the string length. Here's a simpler situation: Consider a tetherball with no windup, like the pole is mounted on a bearing or something. It will rotate freely forever with no change in energy E or angular momentum L (assuming no friction). Now place another pole exactly half the length of the string from the first pole. When the string hits this new pole, the ball will travel in a new circle that intersects the first pole. At first, there is no change in either E or L, but as the ball nears the first pole (imagine it is very small and can pass through it), it will still have the same energy E, but the L (which you will recall is measured from the centre point of the first pole) is exactly zero! How can this be? The answer is that the string applied a force to pole 2 and changed the Earth's rotation.

 

[pervect]

Still stuck, eh? Try answering this simpler question.

An object of mass M is moving up the page at a velocity of V meters per second. A string is pulling on the object to the left with a force of F Newtons.

What is the rate of change of energy of the object? (Hint: the rate of change of energy is called _work_.)

---------------M (moving up) ^

string, pulling M to the left

 

[eosphorus]

i understand you mean that as the radius of the tetherball decreases the momentum of the ball decreases to keep the kinetic energy of the ball constant, that momentum lost by the ball is transferred to Earth via the radius of the pole and the string tension

but if the Earth has now some momentum it didnt have before it means also has some kinetic energy it didnt have before and the ball didnt lose any of its energy, so from where comes this kinetic energy acquired by earth?

i read several times the thread dedicated to the tetherball but you just talked about the inwards tetherball what happens with the outwards one?

if the speed of the ball remains constant the momentum of the system grows with the radius, doesn't this contradict conservation of momentum?

besides the Earth acquire a rotation because of the radius of the pole and the tension of the string, teoretically the tetherball could unwind forever so the torque applied on Earth would tend to infinite, how is this posible?

this are the reasons that make me hard to believe the speed of the ball remains constant, i think the ball should lose more speed as the pole gets thicker because the thicker it is the more rotation the Earth will get from the ball

as for the ball going up being pull to the left by a spring, i don't know i suppose is a simultaneous case of exchange of kinetic energy by potential energy by gravity and a spring

 

[Doc Al]

i understand you mean that as the radius of the tetherball decreases the momentum of the ball decreases to keep the kinetic energy of the ball constant, that momentum lost by the ball is transferred to Earth via the radius of the pole and the string tension

but if the Earth has now some momentum it didnt have before it means also has some kinetic energy it didnt have before and the ball didnt lose any of its energy, so from where comes this kinetic energy acquired by earth?

You are correct! Realize that when we say that the energy (and speed) of the tetherball does not change, that is an approximation that assumes a perfectly fixed pole attached to an infinitely massive earth. In reality, the ball does pull on the pole, making it move ever so slightly--just enough so that the total angular momentum of "tetherball + earth" is conserved as krab explained. So some energy does get transferred from the tetherball to the rotating earth.

To find out how much energy the tetherball loses, just calculate the amount of energy the Earth must have if its angular momentum increases as the conservation of momentum requires. You will find that the amount of kinetic energy gained by the Earth is astronomically small, since the Earth is so massive. Thus our "approximation" that the tetherball does not lose energy is a very good one. (Do this calculation for yourself and see.)

i read several times the thread dedicated to the tetherball but you just talked about the inwards tetherball what happens with the outwards one?

Same thing.

if the speed of the ball remains constant the momentum of the system grows with the radius, doesn't this contradict conservation of momentum?

No. Just like before, the Earth's angular momentum will change so that the total angular momentum is conserved.

besides the Earth acquire a rotation because of the radius of the pole and the tension of the string, teoretically the tetherball could unwind forever so the torque applied on Earth would tend to infinite, how is this posible?

When you talk about "infinities", lots of crazy things can happen. (Add one drop of water to the ocean: no big deal. Now add an "infinite" number of drops: Oops! There goes the galaxy! )

this are the reasons that make me hard to believe the speed of the ball remains constant, i think the ball should lose more speed as the pole gets thicker because the thicker it is the more rotation the Earth will get from the ball

Estimate the increased rotation of the Earth as I suggest above and then you'll get a more realistic understanding.

 

[eosphorus]

thanks but two last questions:

if the momentum of Earth were 0 then shouldn the tetherball going outwards half its speed when doubling the radius?

the inwards tetherball transfers its momentum to Earth via the radius of the pole and the tension of the string, but how does Earth transfer its momentum to the outwards tetherball?

 

[Doc Al]

if the momentum of Earth were 0 then shouldn the tetherball going outwards half its speed when doubling the radius?

I don't understand the question. Do you mean if the angular momentum of the Earth was initially zero?

In any case, the answer is NO.

the inwards tetherball transfers its momentum to Earth via the radius of the pole and the tension of the string, but how does Earth transfer its momentum to the outwards tetherball?

In a similar manner. (The cord will make a different angle with the pole.)

 

[eosphoro]

so if the angular momentum of Earth is 0 and the tetherball gains angular momentum because it doesn't half its speed as the radius doubles where would the tetherball obtained its increased momentum from?

 

[Doc Al]

If the earth/pole exerts a torque on the tetherball, then the tetherball exerts an equal (but opposite) torque on the earth/pole.

Say two astronauts are floating in space next to their ship. Relative to the ship, their momentum is zero. But one guy shoves the other and both go flying in opposite directions. Since they started with zero momentum, where did their momenta come from? Same question.