Angular Momentum

1. Angular Momentum The mass of an object (m) is the measure of its resistance to translational acceleration. The moment of inertia of an object (I) is the measure of its resistance to angular acceleration. How does quantifying I help us to understand objects in rotational motion? (How did quantifying m help us to understand objects in translational motion?) OSU PH 212, Before Class #5

2. Another Fundamental Consistency in the Universe: The Conservation of Angular Momentum The whole point of defining I properly is to understand the “persistence” of angular motion—just as we understand the “persistence” of translational motion: Momentum. P = mv (“inertia times velocity”) for linear momentum. Similarly, we write angular momentum as the product of rotational inertia and angular velocity: L = I* And like its linear counterpart, angular momentum is apparently a property of all matter—every object (anything with mass) in the universe. And, again, like linear momentum, so far as we have observed: The total quantity of this property, angular momentum, in the universe is constant. In other words, it cannot be either created or destroyed. It can only be transferred between objects. *at least for objects symmetric about a rotational axis. OSU PH 212, Before Class #5

3. Here are some of the direct analogies between (linear) translational and rotational motion: Quantity or PrincipleLinearRotation Position x Velocity v Acceleration a Inertia (resistance to mass (m) moment of acceleration) inertia (I) Momentum P= mvL= Iw OSU PH 212, Before Class #5

4. Thus we can argue the same case as with linear momentum: In an isolated system (any set of objects unaffected by any exchanges of momentum except with one another), L = 0. That is, Li = Lf. Q: An ice-skater is standing upright and spinning in place on the ice, with her arms outstretched and no friction to slow her rotation. Then she draws her arms in toward her body—toward her axis of rotation. What must happen? A: There is no external influence on her rotation, so L = 0. But by drawing in her arms, she is reducing her moment of inertia for this rotation: There is more mass (her arms) closer to the axis of rotation. So if L does not change, but I has decreased,  must increase. OSU PH 212, Before Class #5

5. Two buckets spin around in a horizontal circle on frictionless bearings. When it begins to rain (no wind, with vertical rainfall) into the buckets… • The buckets speed up because the gravitational potential energy of the rain is transformed into kinetic energy. • The buckets continue to rotate at constant w, because the rain falls vertically but the buckets move horizontally. • The buckets slow down, because the angular momentum of the bucket + rain system is conserved. • The buckets continue to rotate at constant w, because the total Emech of the bucket + rain system is conserved. • None of the above. OSU PH 212, Before Class #5

6. Two buckets spin around in a horizontal circle on frictionless bearings. When it begins to rain (no wind, with vertical rainfall) into the buckets… • The buckets speed up because the gravitational potential energy of the rain is transformed into kinetic energy. • The buckets continue to rotate at constant w, because the rain falls vertically but the buckets move horizontally. • The buckets slow down, because the angular momentum of the bucket + rain system is conserved; and the rain contributed additional mass but not any additional angular momentum around the buckets’ axis. • The buckets continue to rotate at constant w, because the total Emech of the bucket + rain system is conserved. • None of the above. OSU PH 212, Before Class #5

7. (Similar to Prep 2, item 4b) A figure skater spins in place on frictionless ice. At first, with her arms outstretched, her angular velocity is 10.0 rad/s. Then she draws in her arms, which decreases her moment of inertia about the axis of rotation by 20%. What is her new angular velocity? A. 7.50 rad/s B. 8.00 rad/s C. 12.0 rad/s D. 12.5 rad/s E. None of the above. OSU PH 212, Before Class #5

8. (Similar to Prep 2, item 4b) A figure skater spins in place on frictionless ice. At first, with her arms outstretched, her angular velocity is 10.0 rad/s. Then she draws in her arms, which decreases her moment of inertia about the axis of rotation by 20%. What is her new angular velocity? A. 7.50 rad/s B. 8.00 rad/s C. 12.0 rad/s D.12.5 rad/s (Conservation of L: Iiwi = Ifwf) E. None of the above. OSU PH 212, Before Class #5

9. Can collisions and explosions happen with rotating objects? Yes. Suppose two objects, each able to spin about the same axis, collide and stick together. Or suppose a single mass drops onto a rotating carousel and comes to rest at a point some distance from the hub. (See Prep 2, items 5a and 6c, respectively.) Those are cases of “two objects becoming one”—a collision —and at least one of the objects has rotational motion and therefore angular momentum. But the new combined object has a different I and therefore a different w. Or, what about the opposite process? For example, what if a child begins to walk around a rotating carousel? This is “one object becoming two”—an explosion—and each object now has its own I and w around the axis. See Prep 2, item 6a. OSU PH 212, Before Class #5