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

Engineering

Physics

Variable Mass System

Question

$\frac{1}{2} {λℓv}^{2} + \frac{{λℓ}^{2} g}{2}$

λℓgv

$λℓgv + \frac{1}{2} {λv}^{3}$

${λv}^{3} + \frac{λℓvg}{2}$

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$F = mg + \frac{vdm}{dt} = λyg + {λv}^{2}$

$P_{avg} = \frac{\int Pdx}{\int dx} = \frac{\int Fvdx}{\int dx} = {λv}^{3} + \frac{λℓvg}{2}$