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 vt and vb, and the corresponding tensions in the string are T⃗t and T⃗b. T⃗t and T⃗b have magnitudes Tt and Tb.
Express the difference in tension in terms of m and g. The quantities vt and vb should not appear in your final answer.
Ki+Ui+WNC = Kf+Uf
The Attempt at a Solution
Ki+WNC = Kf+Uf
WNC = Kf+Uf - Ki
WNC = (1/2)mvt2 + mg2L - (1/2)mvb2
Tt = (mvt2/L) - mg
Tb = mg - (mvb2/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.