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7.7: Interesting Effects Involving Angular Momentum

There are two fascinating aspects of angular motion that don’t exist for linear motion in quite the same way. The first is that the rotational inertia is readily changed, as for example, when a skater extends or pulls in a leg. The second is related to the fact that both p and L are vector quantities and can change in direction without changing in magnitude

When an ice skater begins to spin with a leg extended, there is only a small torque exerted on the skater by the ice. Thus, angular momentum diminishes rather slowly (she can spin for a long time). Now, if she pulls in her leg, her rotational inertia is reduced considerably, and her rotational velocity (spin velocity) increases considerably. This is most easily seen by writing the angular momentum as \( L = I\omega\) and noting that if L remains almost constant, then the product \(I\omega\) must remain constant.

Another fascinating, and rather startling situation, is the change in direction of the angular momentum of a spinning object when it is acted upon by a torque that is not along the direction of the angular momentum vector itself. This is the weird behavior exhibited by a spinning top or gyroscope.

Figure 7.7.1 shows a bicycle wheel supported by a rope at the left end of a short axle. Figure 7.7.2 is the extended force diagram. The torque caused by the force of the Earth acting down at the center of gravity of the wheel produces a torque that is perpendicular to this force and the axle; it points into the figure. If the wheel is not spinning, it just falls, rotating about the pivot point, because this is the only point of support. However, when the wheel is spinning with angular momentum \(L_0\), the situation is much different.

Figure 7.7.1                         Figure 7.7.2

Figure 7.7.3 is a top view, showing the original angular momentum vector, \(L_i\), the new angular momentum vector, \(L_f\) and the torque\(\tau\) . The torque acts for a time \(\Delta t\) .

Figure 7.7.3

We use the angular impulse equation to give the change in angular momentum,

\[ \tau \Delta t= \Delta L = L_f = L_i \]

or

\[  L_f = L_i + \tau\Delta t \]

That is, the direction of the initial angular momentum, \(L_i\), is changed by the presence of the angular impulse, and is moved to the direction shown by \(L_f\). But if L is in a new direction, then the orientation of the wheel must have changed, because L is due to the spinning wheel and points along \(\omega\) . This turning motion of the orientation of the wheel is called precession. Instead of falling, the wheel precesses. Of course, once the angular momentum (and the wheel) point in a new direction, the torque comes into play again, causing the wheel to precess still farther. In this fashion, the wheel is caused to precess in a horizontal circle about the pivot point.

Precession is analogous to the situation of a ball being twirled around in a circle on the end of a string. Why doesn’t the tension in the string pull the ball in toward the center of the circle? The answer is that it does, but the large tangential velocity also moves the ball in a direction tangent to the circle. The net result is that the ball travels in a circular path. If there were no large tangential velocity, the ball would indeed be pulled directly toward the center of the circle due to the tension in the string. A similar thing happens with the bike wheel. The torque causes a change in the direction of the large angular momentum of the spinning wheel. If the wheel did not have this large angular momentum, the torque would cause the wheel to tip over, or “fall down.”