Most of us have seen a figure skater whose big spinning fina

Most of us have seen a figure skater whose big spinning finale at the end of her routine makes her into a veritable blur. At first, she\'s spinning at some reasonable rate, but then-with nobody giving her any kind of push to go faster, she\'s able to increase her rate of rotation a LOT. How does do that? Find out. Tab lab is short on ice rinks, but we do have a rotating chair. Have someone from your group-(someone who is not prone to motion sickness) climb aboard the chair. To accentuate the effect, have him/her take one of the weights in each hand and hold out his/her hands and also extend his/her legs straight out. Have everyone else back up, then one other team member give the volunteer a (gentle) turn-not very fast! After that person has finished accelerating the volunteer and has stepped back-assuming the chair\'s axle is has low friction (and ignoring the slow rotation of the earth), the volunteer and chair are essentially an isolated rotating body-like a figure skater spinning on slick ice. Let the volunteer go around a couple of times, then ask him/her to draw in his/her arms and legs...then extend them out again....What happens-and why? Answer by using the Conservation of Angular Momentum: Write an appropriate equation and rearrange it to make your point. (When the volunteer has had enough, be sure to stop the chair.) Now, let another volunteer from the group-someone with good arm strength-put on the gloves and sit in the chair. This time he/she should remain basically still. Then someone else should hold the bicycle wheel so that its axis is vertical and someone should give it a strong spin-get it going pretty good-clockwise, as viewed from above. Now, without changing the orientation of that axis, carefully give the spinning wheel to the person in the chair and stand back. That person should hold the bottom axis in one hand and use the other (gloved) hand to brush against the outside of the spinning wheel (don\'t catch any fingers in the spokes!), until the wheel is at rest against the glove. What happens? Why? Answer by using the Conservation of Angular Momentum: Write an appropriate equation and rearrange it to make your point. Predict: What would have happened if the bike wheel had been going counter clockwise instead of clockwise? Explain your prediction with an equation. Now test your prediction-repeat step b with the bike wheel going counter clockwise (as viewed from above). What happens? Predict: What will happen if the volunteer in the chair, holding the stopped bike wheel (always with its axis vertical) now gives it a strong push to start it turning again? Will direction matter? Explain your predictions with equations. Now test your prediction-once for each spinning direction. What happens? When land ice on Antarctica melts and flows into the sea, eventually it circulates and therefore distributes uniformly over the earth\'s oceans. As a result, does the earth\'s rotational speed increase, decrease or remain unaffected? (Does this change the length of one day? If so, how?) As always, explain your thinking fully.

Solution

For a body rotating about an axis in absence of any external force, angular momentum remains conserved. Same as it happens with linear momentum. Further, we also know that angular momentum about an axis for body is equal to the product of moment of inertia of the body about that axis and angular velocity. That is, J = I x w; where J is angular momentum, I is inertia and w is the angular velocity.

a.) Now, as for the skater making the spin, her arms when stretched out, increases the moment of inertia about an axis, say, passing through the centre of her body [vertically upwards]. Think of it in terms of mass being put at a farther distance. As in, when she sticks her arms out, each particle that constitutes her arms, is placed at a greater distance from the central axis, hence the total inertia for all the particles would be sum of mass X distance from axis^2.

Ii = Mi x Ri^2; here i denotes one particle.

In short, spreading the arms increases the inertia, while pulling them back in reduces the inertia.

Use that for conservation of momentum, in absence of external force, we can see that for same angular momentum, the angular velocity has to be larger for smaller inertia

That is; I1 x W1 = I2 x W2

That is, W2 = (I1/I2) * W1; Therefore, decrease in inertia, that is pulling in the arms, increases the angular velocity.

b.) Now, for the volunteer sitting on a chair with a spinning wheel and trying to stop it, the angular momentum for the system will remain conserved in absence of external force, that is, by the time the wheel stops with respect to the volunteer, the volunteer himself [the entire system of volunteer + chair + wheel] will start spinning with respect of others. Same equation as above will apply.

Analogy: Think of it in terms of linear momentum; you try and stop moving object while sitting on chair. Because of the momentum of the particle moving, you would get \'pulled\' along with it. In other words, you would start moving in the same direction as the particle. Similar thing happens in the above mentioned situation.

c.) This part is same as above with direction changed.

d.) Here, the still volunteer is trying to rotate the wheel himself. Now, in absence of any external force, overall angular momentum as to be conserved, which happens to be zero in this case, as the system was still inititally.

Hence, 0 = I1 X W1 + I2 X W2; here the I1XW1 is the angular momentum of the wheel while I2XW2 is the angular momentum of the volunteer + chair.

That is, W2 = - (I1/I2) x W1 [Minus denotes opposite direction]

Therefore, the volunteer will start spinning in a direction opposite of the wheel.

e.) The axis of rotation of the earth passes through the poles and the point on earth farthest from the axis is on the equator.

Hence, if we melt the ice caps and distribute the water around the surface of the earth, it would be more like moving some mass away from the poles (closer to axis) to the equator and other areas (away from the axis)

By the same reasoning as above, the inertia of the earth will increase, hence the rate of rotation will decrease as the angular momentum should remain conserved.