Does Tension Do Any Work in Uniform Circular Motion?

[Heat]

Homework Statement

A ball of mass 0.765 kg is tied to the end of a string of length 1.59 m and swung in a vertical circle.

During one complete circle, starting anywhere, calculate the total work done on the ball by the tension in the string.

During one complete circle, starting anywhere, calculate the total work done on the ball by gravity.

Repeat part (a) for motion along the semicircle from the lowest to the highest point on the path.

Repeat part (b) for motion along the semicircle from the lowest to the highest point on the path.

Take all free fall acceleration to be 9.8 m/s^s

Homework Equations

W = F * s
K = .5mv^2

The Attempt at a Solution

For the first part,

I tried getting the C, by using C = 2pi1.59^2
c = 9.99
then 9.8(9.9) = 97.02, but it's incorrect :(

 

[learningphysics]

What is the definition of work? Hint... the tension at any time is perpendicular to the displacement at that particular time...

 

[m2003]

Dear student,

Thank you for providing the necessary information and equations for this problem. Based on the given information, we can approach this problem in the following way:

1. Tension Work: In order to calculate the total work done on the ball by the tension in the string, we need to first understand the work-energy principle. This principle states that the work done on an object is equal to the change in its kinetic energy. In this case, the ball is moving in a circular motion, so its kinetic energy will be changing as it moves from one point to another.

To calculate the total work done by the tension in the string, we need to consider the work done at each point in the circle. At any given point, the tension in the string is providing the centripetal force that keeps the ball moving in a circular path. This force is given by F = mv^2/r, where m is the mass of the ball, v is its velocity, and r is the radius of the circle (in this case, the length of the string).

Since the ball is moving in a vertical circle, its velocity will be changing as it moves from the bottom to the top of the circle. At the bottom of the circle, the velocity will be at its maximum (vmax) and at the top of the circle, the velocity will be at its minimum (vmin). Therefore, the work done by the tension at the bottom of the circle will be equal to Fbottom * s, where Fbottom = mvmax^2/r and s is the distance traveled (which is equal to the circumference of the circle, 2πr). Similarly, the work done by the tension at the top of the circle will be Ftop * s, where Ftop = mvmin^2/r.

Since the ball is completing one full circle, we need to add the work done at both the bottom and top of the circle to get the total work done by the tension. Therefore, the total work done by the tension can be calculated as follows:

Wtension = Fbottom * s + Ftop * s = (mvmax^2/r) * 2πr + (mvmin^2/r) * 2πr = m(vmax^2 + vmin^2) * π

2. Gravitational Work: In order to calculate the work done by gravity, we need to consider the change in the potential energy of the ball as it