Finding Centripetal Force in a Closed-Loop String

[dragonlorder]

Homework Statement

I never thought I would have this kind of elementary problem
consider a string closed-loop spinning around an axis, and its shape is a circle, I wanted to find the centripital force at each point. (uniform density is assumed). I have problem expressing the mass

Homework Equations

The Attempt at a Solution

I apply F=ma , and F=m f(angle), the acceleration is obviously a function of the angle. But how do I write the mass at a point of a continuous string. The formula F=m f(angle) works for 1000, or 10000 mass points spinning around an axis, since the mass is given for each mass point, but what about a string, its continuous. But if I write dm (differential form), it would distort the formula...

 

[kuruman]

If you have a continuous string, you can pick a point on the loop and consider a piece of arc length ds at that point. Then if the mass of the loop is M and its radius R, the mass of your piece is

dm = Mds/2πR

Once you have this, the centripetal force on that element is

dF = dm ω²r

where ω is he instantaneous angular speed and r the distance of your element of mass dm to the axis of rotation.

Is this what you had in mind?

 

[dragonlorder]

If you have a continuous string, you can pick a point on the loop and consider a piece of arc length ds at that point. Then if the mass of the loop is M and its radius R, the mass of your piece is

dm = Mds/2πR

Once you have this, the centripetal force on that element is

dF = dm ω²r

where ω is he instantaneous angular speed and r the distance of your element of mass dm to the axis of rotation.

Is this what you had in mind?

yea, but I feel weird having to integrate dF. The formula works for finite masses, but as the number goes infinite, it fails. But the formula for continuous rope should have the form similar to that of finite. that's why I feel weird to integrate dF.

 

[tms]

yea, but I feel weird having to integrate dF.

Why do you want to integrate? The question, as you posted it, asks for the force at each point, not the total force.

The formula works for finite masses, but as the number goes infinite, it fails.

If you do integrate, the mass will be a constant; you will integrate over ds.

 

[dragonlorder]

Why do you want to integrate? The question, as you posted it, asks for the force at each point, not the total force.


If you do integrate, the mass will be a constant; you will integrate over ds.

thats exactly the problem I am having. how to modify that formula for the continuous situation

 

[tms]

thats exactly the problem I am having. how to modify that formula for the continuous situation

You have already been given the main part of the answer.