# Need help with relationship between angular momentum, linear and angular 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

## 3 Answers

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.

• Your answer to "4) Then why would something like a pulsar rotate faster as its matter get closer to the center?" is not correct, imo. Before the reunion the matter that is being accreted is rotating much slower than the pulsar and has to be accelerated upon reunion. Thus the momentum of the pulsar has to be shared and the pulsar have to slow down. In the ice skater the arms are already rotating at the same speed. Think as if you are the motor of a rotating carrousel and 100 people jumped inside at once and their velocity had to go from 0 to... Dont you think that they will slow you down? Apr 13, 2018 at 0:19

Think of a rotating disk. The disk is rotating on its center 1. Since the angular velocity, as you've mentioned, is the rate of change of angular position, then every point on the disk will experience the same angular velocity. If the disk change in shape as it rotate, then the angular velocity might not be the same anymore. 3. yes, you're right. since the linear velocity is the arc length that is traveled per unit time and this arc length is depend on the distance of the point to the center of the disk, the linear velocity will decrease as it closer to the center of the disk. at the center of the disk, the linear velocity will be zero. Imagine that we put a small cube at the center of the disk while the disk is rotating. That cube will be rotating as well (move in circular motion), but it doesn't travel any distance. I hope that'll help a bit

Rotational inertia is the resists any change of rotational velocity If you decrease rotational inertia,angukar velocity will increase The reason why when the distribution of mass is closer to the axis of rotation rotational inertia decreases is that the magnitude of rotational inertia is proportional to the perpendicular distance betwen a mass element and the axis of rotation which is denoted as R I=mR^2 So when mass distribution is closer to the axis of rotation , R will decrease and therefore rotational inertia I will also decrease The angular momentum of an isolated system is conserved or a constant which means that net angular momentum Lnet = Li+Lf =0 -> Lf= Li or Iiwi =If*wf Where Li is net angular momentum at time Ti, Lf is net angular momentum at time Tf, Ii is rotational inertia at time Ti, If is rotational momentum at time Tf, w = angular speed subscript i stands for initial , f stands for final So if Rotational momentum of a later time Tf decreases , the angular speed at that time Tf must increase in order for the net angular momentum Lf at time Tf to be still equal to the net angular momentum at earlier time Ti because the system's net angular momentum is a constant it has to be the same at all points of time