# Question #20a23

Nov 6, 2016

Given

$h \to \text{the depth of the well} = 12 m$

This will also be the length if the rope

$M \to \text{the mass of bucket+ water in it} = 15 k g$

$W \to \text{the work done to raise the bucket}$
$\text{filled with water} = 2160 J$

Let the mass of the rope be m kg

The total work done against the gravitional pull will be

$W = M g h + m g \cdot \frac{h}{2}$

Here $m g \cdot \frac{h}{2}$ is the work done to raise the center of gravity ( at $\frac{h}{2}$ depth) of the rope.

So

$M g h + m g \cdot \frac{h}{2} = 2160$

$\implies 15 \cdot 9.8 \cdot 12 + m \cdot 9.8 \cdot \frac{12}{2} = 2160$

$\implies m = \frac{2160 - 15 \cdot 9.8 \cdot 12}{9.8 \times 6} = 6.73 k g$

So linear density of the rope

$= \text{mass of the rope (m)"/"length of the rope (h)}$

$= \frac{6.73}{12} \approx 0.56 \text{ kg/m}$