A homogeneous rope of mass per unit length ' λ ' and length ℓ kept on ground and one end of the rope is fixed to ground at O. The left end of the rope (with respect to fixed end) is pulled b

Question

A homogeneous rope of mass per unit length ' $λ$ ' and length $ℓ$ kept on ground and one end of the rope is fixed to ground at O. The left end of the rope (with respect to fixed end) is pulled by an external agent which imparts constant velocity V to it. Find the work done by the external agent (in joule) to place the moving end extremely left with respect to fixed end. Take $λ = 1 kg / m, V = 1 ms^{- 1} & ℓ = 1 m$ . (Height of heap is very very small)

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TestHub · Accepted Answer

As the rope leaves the heap at speed V, length is recruited at that speed.
Rate of mass pickup:
$\frac{d m}{d t} = λV$
Momentum supplied per unit time:
$F = V \frac{d m}{d t} = λ V^{2}$
The pulling end travels the rope length:
$W = F ℓ = λ ℓ V^{2}$
With the printed values:
$W = 1 \times 1 \times 1^{2} = 1 J$

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