Circling Ball A ball of mass m is attached to a string of length L . It is being

Question

Circling Ball

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 andvb, and the corresponding tensions in the string are T? t and T? b. T? t and T? b have magnitudes Tt and

Part A

Find Tb?Tt, the difference between the magnitude of the tension in the string at the bottom relative to that at the top of the circle.

Express the difference in tension in terms of m and g. The quantities vt and vb should not appear in your final answer.

Explanation / Answer

First write the centripetal force eqs at top & bottom.
Choose "in" toward the center as positive;
At Top;
Tt + mg = mvt^2/R

At Bottom;
Tb - mg = mvb^2/R

Subtract
Tb - Tt = 2mg + (1/R)[mvb^2 - mvt^2]

Rewrite the last term using conservation of energy;
(1/2)mvb^2 = (1/2)mvt^2 + 2mgR

mvb^2 - mvt^2 = 4mgR

Sub this into the tension eq;
Tb - Tt = 2mg + 4mg = 6mg

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