# Thermal Energy from Friction on a Rope

Question:

A capstan is a rotating drum or cylinder over which a rope or cord slides to provide a great amplification of the rope's tension while keeping both ends free. Since the added tension in the rope is due to friction, the capstan generates thermal energy.

If the difference in tension ($T_0-T'$) between the two ends of the rope is $520N$ and the capstan has a diameter of $10.0 cm$ and turns once in $0.90s$, find the rate $P_{thermal}$ at which thermal energy is being generated.

My Work:

$F = 520N$

$d = 0.1m$

$\Delta T = 0.90s$

$\Delta U = Q - W \implies Q = \Delta U + W$

$W = \tau \Delta \theta = Fd(2\pi) \implies W = 326.7J$

$P = \frac{\Delta W}{\Delta T} = 363W$

However, this is the incorrect answer. (I don't know the correct one.) Can anybody give me any guidance here?

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Maybe you should use radius $r = d/2$ instead of diameter $d$?