Solved Examples of Conservation of Angular Momentum: Rotating Systems

📚 Quick Study Guide

🔍 Angular Momentum: A measure of an object's rotation. For a point mass, it's given by $L = I\omega$, where $I$ is the moment of inertia and $\omega$ is the angular velocity. 💡 Moment of Inertia: Represents the resistance to rotational motion. It depends on the object's mass distribution. For a point mass $I = mr^2$, where $m$ is the mass and $r$ is the distance from the axis of rotation. 📝 Conservation Law: In a closed system, the total angular momentum remains constant if no external torque acts on it. Mathematically, $L_i = L_f$ (initial angular momentum equals final angular momentum). ⚛️ Rotating Systems: When dealing with rotating systems, the total angular momentum is the sum of the individual angular momenta of all parts of the system. ➕ Problem Solving Tip: Identify the initial and final states of the rotating system. Calculate the initial and final angular momenta. Apply the conservation of angular momentum principle: $I_i\omega_i = I_f\omega_f$.

Practice Quiz

1. A spinning skater pulls their arms inward. What happens to their angular speed?

   1. Decreases
   2. Remains constant
   3. Increases
   4. Becomes zero

2. A merry-go-round is rotating with a child standing at the center. If the child moves to the edge, what happens to the merry-go-round's angular speed?

   1. Increases
   2. Decreases
   3. Remains the same
   4. Stops rotating

3. A disk is rotating freely. If a piece of clay is dropped onto the disk, sticking to it, what happens to the angular speed of the disk?

   1. Increases
   2. Decreases
   3. Remains the same
   4. First increases, then decreases

4. Two identical disks are rotating with the same angular speed but in opposite directions. They are brought into contact. What is their final angular speed?

   1. Twice the initial speed
   2. The same as the initial speed
   3. Half the initial speed
   4. Zero

5. A student sits on a rotating stool holding a spinning bicycle wheel. If the student flips the wheel over, what happens?

   1. The student spins in the same direction
   2. The student spins in the opposite direction
   3. The student doesn't move
   4. The wheel stops spinning

6. A rod of length $L$ and mass $M$ is rotating about its center. What is its moment of inertia?

   1. $\frac{1}{2}ML^2$
   2. $\frac{1}{3}ML^2$
   3. $\frac{1}{12}ML^2$
   4. $ML^2$

7. A system consists of a rotating platform and a person standing at the center. When the person walks to the edge, what is conserved?

   1. Kinetic energy
   2. Angular velocity
   3. Angular momentum
   4. Moment of inertia