Question

The mass of bucket full of water is 15 kg . It is being pulled up from a 15 m deep well. Due to a hole in the bucket 6 kg water flows out of the bucket at a uniform rate. The work done in drawing the bucket out of the well will be $(g=10m/s^2)-$

• A. 900 J
• B. 1500 J
• C. 1800 J
• D. 2100 J

Answer 1

The correct option is C 1800 J

$\text{Step 1: Water in bucket at height x}$

Mass of bucket full of water $=15kg$

Water goes out of bucket $=6kg$

Out flow of water per meter of pull $=\frac{6kg}{15m}$

$=0.4kg/m$

Water in the bucket at height x from bottom $\left(m\right)$ $=(15-0.4x)kg$

$m=(15-0.4x)$

$\text{Step 2: Work done}$

Work done in pulling the bucket by small distance $\left(dx\right)$

$dW=mgdx$

$dW=(15-0.4x)gdx$

$\text{Step 3: Integration}$

For total work done integrate from $x=0$ to $x=15m$

$W={\int }_0^{15}(15-0.4x)gx$

$\Rightarrow W=g{[15x-0.4\frac{x^2}{2}]}_0^{15}$

$\Rightarrow W=10(225-45)$

$\Rightarrow W=1800J$

Option C is correct.

$\text{Alternate Solution:}$

Here work done by gravity is linear function of height $\left(x\right)$

So we can find average water leak in this case-

Average weight leak $=3$ g Kg - wt

So remaining $=(15-3)g=12$ g Kg - wt

Work done $=F\times d$ $=12g\times 15$ $=120\times 15$

$W=1800J$

$\text{Note:}$

This method is only applicable for the linear function of height.

If work done by gravity is a non-linear function of $x$ then, don't use this approach

e.g. If $W=mg{x}^3$

Here we can not apply this method