A uniform rope of linear mass density $λ$ and length l is coiled on a smooth horizontal surface. One end is pulled up with constant velocity v. Then the average power applied by the external agent in pulling the entire rope just off the ground is

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$\frac{1}{2} λ v^{2} + \frac{λ l^{2} g}{2}$

$\frac{λ l v g}{2} + \frac{λ v^{3}}{2}$

$λ l g v + \frac{1}{2} λ v^{3}$

$λ l g v$

answer is C.

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Detailed Solution

$K E_{1} = 0, P E_{1} = 0$

$K E_{f} = \frac{1}{2} (λ l) v^{2}, P E_{f} = (λ l) g \frac{l}{2}$

$t = \frac{l}{v}, P a v e = \frac{N e t c h a n g e i n e n e r g y}{t i m e}$ $= \frac{\frac{1}{2} λ l v^{2} + \frac{λ \lg l}{2}}{l / v} = \frac{1}{2} λ v^{3} + \frac{λ l v g}{2}$

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