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## Homework Statement

A ball of mass m is attached to a string of length L. It is being swung in a vertical circle with enough speed so that the string remains taut throughout the ball's motion.(Figure 1) Assume that the ball travels freely in this vertical circle with negligible loss of total mechanical energy. At the top and bottom of the vertical circle, the ball's speeds are v

_{t}and v

_{b}, and the corresponding tensions in the string are T⃗

_{t}and T⃗

_{b}. T⃗

_{t}and T⃗

_{b}have magnitudes T

_{t}and T

_{b}.

Express the difference in tension in terms of m and g. The quantities v

_{t}and v

_{b}should

*not*appear in your final answer.

## Homework Equations

K

_{i}+U

_{i}+W

_{NC}= K

_{f}+U

_{f}

## The Attempt at a Solution

K

_{i}+W

_{NC}= K

_{f}+U

_{f}

W

_{NC}= K

_{f}+U

_{f}- K

_{i}

W

_{NC}= (1/2)mv

_{t}

^{2}+ mg2L - (1/2)mv

_{b}

^{2}

Top:

T

_{t}= (mv

_{t}

^{2}/L) - mg

Bottom:

T

_{b}= mg - (mv

_{b}

^{2}/L)

I don't know what to do after this. I have work done by non conservative forces, but I'm not sure how to relate this to the difference in tensions.