How Do You Calculate Tension in a Spinning String Loop?

[david456103]

Homework Statement

A piece of string of length l and mass M is fastened into a circular loop and set spinning about the center of a circle with uniform angular velocity omega. Find the tension in the string. Suggestion: Draw a force diagram for a small piece of the loop subtending a small angle delta theta.

Homework Equations

The Attempt at a Solution

I know I'm supposed to use differentials and then integrate, but isn't the tension in the rope the same everywhere?

 

[HallsofIvy]

If you mean the string, then, yes, it is.

 

[david456103]

but how do you find that tension?

 

[Redbelly98]

Did you try drawing the force diagram, as suggested? What are the forces acting on the small piece of loop?

 

[paranormal]

Hi there! I would be happy to help you with this Newton's law problem. First, let's start by drawing a force diagram for a small piece of the loop. As you suggested, we will consider a small angle delta theta.

We know that the tension in the string is the force that keeps the string in a circular motion. Therefore, we can draw two forces acting on the small piece of the loop: tension T and centripetal force Fc. These two forces must be equal in magnitude and opposite in direction in order for the string to maintain its circular motion.

Now, we can use Newton's second law to relate these forces to the mass and acceleration of the small piece of the loop. The mass of the small piece can be approximated as dm = (M/l) * delta theta, where M is the mass of the entire string and l is its length. The acceleration can be approximated as a = (omega)^2 * r, where r is the radius of the loop.

Plugging these values into Newton's second law, we get T = (M/l) * delta theta * (omega)^2 * r. We can then integrate this expression over the entire loop to find the total tension in the string.

I hope this helps! Let me know if you have any further questions.