I Conservation of angular momentum in the iceskater example

[renobueno4153]

TL;DR

If an iceskater pulls their limbs together while they are spinning, they speed up. Where does the acceleration come from?

Today in my physics lecture we looked at the conservation of angular momentum.

I dont understand the iceskater example: If an iceskater pulls their limbs together while they are spinning, they speed up.

At first I thought I understand the math behind it but the more I think about the math the more confused I get xD.

What I understand is the case is shown by the conservation of angular momentum meaning the angular momentum is constant. So if the position vector which points outward is reduced the angular momentum has to increase. (L = vector r crossprod vector p).

Then I thought about how the Iceskater pulls their arms in which would create a force inwards since an acceleration is only caused by an applied Force (F=ma). So I thought the inward force would increase the acceleration but then I thought about how when it comes to a force changing a rotational acceleration then we talk about torque. But if a Force is inward to the position vector (lever arm) then the torque is 0. So something huge is off here :3

Then I was thinking how I could reason an acceleration without math and that didnt work. Like I can see how burning petrol ends in a chemical reaction where the energy in the bonds are released but seeing an increase of speed by just shortening a distance seems like hocus pocus.

Hope I could show you guys where my confusion lies. Please correct me for incorrect trains of thoughts. Today I tried to ask in the lecture but my prof just said it had to be like that since the angular momentum is conserved. ;(

 

[anuttarasammyak]

The angular momentum before and after the change in the moment of inertia is conserved:
$$M=I_1\omega_1=I_2\omega_2$$
Assuming $I_1 > I_2$ , it follows that $\omega_1<\omega_2$.
As for the rotational energy,
$$E_1=\frac{M^2}{2I_1} < \frac{M^2}{2I_2} =E_2$$
Therefore, the skaters must exert energy to pull in their arms, which increases the total energy of the system.
M is conserved therefore torque is zero as you considered.

 

[PeroK]

TL;DR: If an iceskater pulls their limbs together while they are spinning, they speed up. Where does the acceleration come from?

There are two aspects to this question. If we accept the conservation of AM (angular momentum), then it is a simple calculation. You might still ask what are the forces on each part of the iceskater's body?

We can look at the forces and acceleration expressed in polar coordinates, which express the motion in terms of radial and tangential (rather than Cartesian) components. In which case, we have a more complicated relationship between the components. If you do the maths, you find the total acceleration vector is given by:
$$\vec a = (\ddot r - r\dot \theta^2) \hat r + (r\ddot \theta + 2 \dot r \dot \theta)\hat \theta$$If we have a force in the $\hat r$ direction, then this requires that the acceleration in the $\hat \theta$ direction is zero, which means that:
$$r\ddot \theta + 2 \dot r \dot \theta = 0$$Note that if $r$ is reducing, then $\dot r \ne 0$. And, if there is already angular motion ($\dot \theta \ne 0$), then the second term is non zero. This implies that $\ddot \theta \ne 0$. And, in fact, we have an angular acceleration of:
$$\ddot \theta = -\frac{ 2 \dot r \dot \theta}{r}$$In particular, if $r$ is decreasing, then $\dot r < 0$ and $\ddot \theta > 0$, which means that the rotation is speeding up.

In conclusion, even if a force is entirely in the radial direction, it may result in acceleration in the tangential direction. This is something that you have to come to terms with physically and mathematically.

 

[Lnewqban]

I dont understand the iceskater example: If an iceskater pulls their limbs together while they are spinning, they speed up.

That inwards movement of the limbs reduces the total angular inertia of the ice-skater (the centers of masses of the arms acquire less radius of rotation each).

Please, see:
http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html

Please, note that the angular inertia or moment of inertia of a rotating body depends on the square of the radius; therefore, any change in the radius induces a greater change of the moment of inertia.

Comparing this case to any linear movement, that would be equivalent to reducing the linear inertia of a moving body by reducing its mass.

 

[PeroK]

Comparing this case to any linear movement, that would be equivalent to reducing the linear inertia of a moving body by reducing its mass.

If a chunk falls off your car, the rest of the car does not necessarily speed up.

 

[A.T.]

Where does the acceleration come from?

Which "acceleration"?

Linear speed changes, because the net force has a non-zero component parallel to velocity.

As for angular speed change without torque, consider a much simpler case: A point mass is moving at constant speed along a straight line. Pick some reference point that is not on that line. No forces are acting so the torque is zero, but the angular velocity around that point changes all the time (as does the moment of inertia).

 

[renobueno4153]

Thank you guys for all the explanations. I think I see it a bit better now. But I still dont see one thing. I think i can see it that the math says that the angular velocity has to increase if the moment of inertia is reduced. But I fail to makes sense if I try to visualize why and where the speeding up happens. Like if i hit the gas pedal in the car, the car accelerates. If I jump my legs exert pressure against the floor and I have an accelerations upwards. But now im thinking about an already rotating system speeding up and I dont have any clue why it would speed up. Do I need to live with the fact that the math adds up? :(

 

[jbriggs444]

Thank you guys for all the explanations. I think I see it a bit better now. But I still dont see one thing. I think i can see it that the math says that the angular velocity has to increase if the moment of inertia is reduced. But I fail to makes sense if I try to visualize why and where the speeding up happens. Like if i hit the gas pedal in the car, the car accelerates. If I jump my legs exert pressure against the floor and I have an accelerations upwards. But now im thinking about an already rotating system speeding up and I dont have any clue why it would speed up. Do I need to live with the fact that the math adds up? :(

As the skater's hands are pulled in, they trace out a spiral trajectory. A purely radial force has a non-zero component along the direction of the spiral path. The hands speed up accordingly.

If the hands end up moving too fast to keep pace with the skater's body (as they must) then the body will slow them down and the associated torque will speed the body's rotation up so that hands and body continue to keep pace.

If one shifts to a rotating frame of reference, the same behavior can be described using the Coriolis force.

 

[A.T.]

But now im thinking about an already rotating system ...

Then don't think about a 'rotating system', because the ice skater is not an example of simple rigid body rotation, and such rotation is not required to have angular momentum conservation. Try to understand the same issue in the simpler linear motion example:

A point mass is moving at constant speed along a straight line. Pick some reference point that is not on that line. No forces are acting so the torque is zero, but the angular velocity around that point changes all the time (as does the moment of inertia).