[Physics] Need help with relationship between angular momentum, linear and angular velocity

angular momentumangular velocity

I am in an introduction to engineering physics course and just trying to see if my understanding of angular motion is correct or if I have the wrong idea. So as I understand it, angular velocity is the rate of change of angular position, not physical distance traveled. Arc length is the physical distance traveled, and the rate of change of the arc length is the linear velocity, which depends on the distance the rotating particle is from the axis of rotation, and the rate of change of the angular position.

Questions about this:

  1. Does this mean that for any particle on the rotating body the angular velocity is the same?
  2. Does this mean that when angular momentum is described, we are technically still describing a relationship between linear velocity and mass (mv), only now the linear velocity depends on angular velocity and radial distance from axis?
  3. this would mean that linear velocity would be less for particles close to the axis of rotation, but angular velocity would be the same?
  4. Then why would something like a pulsar rotate faster as its matter get closer to the center?

Sorry I have a lot of questions, but I would be very grateful is someone could clear any of them up

Best Answer

1) Does this mean that for any particle on the rotating body the angular velocity is the same?

On a rigid rotating body, yes, the angular velocity is the same for every point in that body.

2) Does this mean that when angular momentum is described, we are technically still describing a relationship between linear velocity and mass (mv), only now the linear velocity depends on angular velocity and radial distance from axis?

In effect, yes. What you are setting up is an equation of momentum for every infinitesimal mass element of your body. You see the analogy between linear and angular momentum: $$p = mv$$ and $$L = I\omega$$ where $I$ depends on the distribution of mass, not just on the total mass itself.

3) this would mean that linear velocity would be less for particles close to the axis of rotation, but angular velocity would be the same?

That's exactly what's happening. To visualize this, simply imagine spinning a weight fixed to a string over your head. If you spin one weight with a certain angular speed $\omega$ and then release the string, it will fly off at a certain speed. Do the same now with a shorter string but the same angular speed. The weight will fly off at a slower velocity.

4) Then why would something like a pulsar rotate faster as its matter get closer to the center?

Since angular momentum is conserved, decreasing the moment of inertia increases angular speed: $L = I \omega$. As an analogy, consider a pirouette of an ice skater. If the ice skater has her arms outstretched (big moment of inertia) and rotates at a certain angular speed $\omega$, after she pulls in her arms, she will spin at a faster rate. This is because the angular momentum she had before is the same as afterwards.

All in all, angular speed/momentum follows a neat analogy with linear speed/momentum.

Related Question