# Proper description of uniform circular motion

"Proper" description of uniform circular motion

## Homework Statement

Hey guys, the world's a great place because I just finished my archeology term paper. ...so now I have to focus on this physics.

I was just wondering if something's correct. Say I have a mass, like a puck, moving around on the end of a (massless) string in uniform circular motion on a frictionless surface, and this string goes through a hole in the surface (at the center of the circle of motion made by the puck) and is attached to a hanging mass. The hanging mass is motionless.

Trying to describe the position of the puck (looking down on it from above), I wrote

s(t) = rcos((v/r)t)i + rsin((v/r)t)j

(with radius r, speed v and time t)

The magnitude of this is r, the magnitude of its derivative is v, and the magnitude of its second derivative is (v^2)/r, which is centripetal acceleration.

My question is whether this would be "proper." I ask this because 1) using the derivative of this for the velocity equation would certainly properly describe the velocity at any time t, but it would do so in terms of the magnitude of the velocity, and that seems like cheating.

I'm sorry if this isn't the proper forum for this question (this question is for homework, though). What do you think? Is saying that the velocity of an object in uniform circular motion is just [[ the magnitude of the velocity times -sin((v/r)t) for one dimension and cos((v/r)t) for another ]] like saying one equals one times one?

Is, perhaps, v/r relateable to other values in this problem?

Thankz, thx, and thank you.

## The Attempt at a Solution

CompuChip
Homework Helper

v/r is the angular velocity (also denoted by small omega or in plain-text by w :tongue:). Usually, this is constant (like on a frictionless surface, where no external forces act). What you get by deriving s(t) is the linear velocity v(t) which will also have a direction. Yes, this can be expressed in terms of w, and hopefully you will get that its magnitude is always w * r.

In fact, you can generally write
s(t) = r cos(w(t) t)i + r sin(w(t) t)j
where w(t) has an a priori non-trivial time dependence and show that |v(t)| = w(t) * r always holds, relating angular and linear velocity.

Actually, v(t) is a rather useless quantity in general, since w(t) already contains all the information. Only in questions like: "in what direction will the puck fly off if the string is cut" is it actually important.

I think where you are getting confused is that v is NOT the derivative of r. They are in different directions: r is a radial position, while v is a tangential velocity. Like CompuChip said, v and r are related by omega, not by a time derivative.

-Kerry

CompuChip