# Velocity of C of M of Rotating Object = R • w

In the formula for velocity of a rotating object’s center of mass, the radius is multiplied by the angular velocity. I understand why angular velocity and radius are related, as in why they’re put into the same formula, but why are they multiplied? Why are they not added? I also understand they can’t be added because of dissimilar units, but I’m more looking for a physical interpretation. What does it mean physically when angular velocity and radius are multiplied?

• surely you mean that, because they have different units, they cannot be added. May 13 '18 at 2:11
• In general the velocity of the centre of mass of a rotating object is not its angular velocity times its radius. eg a roundabout in a playground. Perhaps you are thinking of a ball or disk which rolls without slipping. You ought to explain the context of your question. May 14 '18 at 10:06
• Sorry, I figured giving the formula would be enough to explain the premise of my question. May 16 '18 at 1:16

## 2 Answers

This is going to be similar to Javatasse's answer, but I thought I'd put a few more explicit formulas. Consider for instance the common example of a carousel. Say you're sitting on one of the horses a distance $r$ from the center of the carousel, and it takes a time $T$ to make one full revolution so that your horse has returned to its starting point.

In making this round trip, you traveled through an angle of $2\pi$ radians in a time $T$, so your angular speed $\omega$ is defined by $$\omega=\frac{2\pi}{T}$$ Now what about the distance you travel? The circumference of the circle you traveled along is given by $2\pi r$, which you have also traveled in time $T$. Therefore, your tangential speed $v$ is given by $$v=\frac{2\pi r}{T}$$ Comparing these two equations, we see that $$v=\omega r$$

• I'm glad it helped! Please consider accepting it as an answer if you found it sufficient.
– Luke
May 13 '18 at 2:20

The velocity is determined by the ratio distance / time. In a circular motion the distance has the relation 2r*pi to the radius. If you multiple either the r or the w the velocity is also multiplied by the same factor.

Examples

• You triple the radius, but keep the angular speed. This means the roation takes the same time, but the distance is 3 times as long.

• You double the angular speed. In this case the object will make two orbits in the same times as it did with one before.

Also you should keep in mind that you can add only 2 things of the same type. An apple plus a pear remains an apple and a pear.

In physics formulas usually express a relation between different measures.