Question

How do you show that the angular momentum of a system is conserved if both the velocity (V) and the angular velocity (w) are constant in time?

Answer 1

To show that the angular momentum of a system is conserved when both the linear velocity (V) and the angular velocity (ω) are constant in time, we can use the following approach:

1. Start with the definition of angular momentum (L) for a point particle or an extended body:

   L = Iω

   Here, I represents the moment of inertia of the object and ω is the angular velocity.

2. If both V and ω are constant in time, the linear velocity V is related to the angular velocity ω by the equation:

   V = Rω

   Here, R represents the distance between the rotation axis and the point at which the velocity is measured.

3. Express the moment of inertia I in terms of the mass (m) and the distance R:

   I = mR²

   This equation holds for a point particle. If you have an extended body, you would need to use the appropriate moment of inertia for that body.

4. Substitute the expression for I into the equation for angular momentum:

   L = Iω = mR²ω

5. Since both V and ω are constant, we can rewrite ω as V/R:

   L = mR²( V/R) = mVR

6. Now, we need to show that L is constant with time. To do that, consider the time derivative of L:

   dL/dt = md(VR)/dt

7. Since V and R are both constants, their derivatives with respect to time are zero:

   dL/dt = m(dV/dt)R + mV(dR/dt)

   But dV/dt = 0 and dR/dt = 0, so the entire expression becomes zero:

   dL/dt = 0

8. Since the time derivative of angular momentum is zero, we conclude that angular momentum (L) is conserved in this system.

Therefore, if both the linear velocity (V) and the angular velocity (ω) are constant in time, the angular momentum of the system remains constant or conserved.