Proper description of uniform circular motion

In summary, the conversation discusses the proper description of uniform circular motion and the relationship between linear and angular velocity. The position of the object in motion can be described using s(t) = rcos((v/r)t)i + rsin((v/r)t)j, with radius r, speed v, and time t. The magnitude of the derivative of this equation is equal to the linear velocity, which is always equal to the angular velocity (denoted by w or omega) multiplied by the radius. The conversation also addresses the confusion between r and v, emphasizing that they are in different directions and not related by a time derivative.
  • #1
Oijl
113
0
"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.

Homework Equations


The Attempt at a Solution

 
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  • #2


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.
 
  • #3


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
 
  • #4


Ah, I failed to stress that point.
As KLoux points out: r is the magnitude of s, which is constant, just like v, which is the magnitude of s'.
 

What is uniform circular motion?

Uniform circular motion is a type of motion in which an object moves along a circular path at a constant speed. The object's velocity remains constant, but its direction is constantly changing.

What are the components of uniform circular motion?

The two components of uniform circular motion are tangential velocity and centripetal acceleration. Tangential velocity is the speed at which the object moves along the circular path, while centripetal acceleration is the acceleration towards the center of the circle.

What is the equation for calculating tangential velocity in uniform circular motion?

The equation for tangential velocity in uniform circular motion is v = rω, where v is the tangential velocity, r is the radius of the circular path, and ω is the angular velocity (the rate of change of the object's angular position).

What is the centripetal force in uniform circular motion?

The centripetal force is the force that keeps an object moving in a circular path. It is always directed towards the center of the circle and its magnitude is equal to the mass of the object multiplied by the square of its tangential velocity divided by the radius of the circle.

What is the difference between uniform circular motion and non-uniform circular motion?

In uniform circular motion, the object's speed remains constant while its direction changes, whereas in non-uniform circular motion, the object's speed and direction both change. Additionally, the magnitude of the centripetal force may vary in non-uniform circular motion.

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