Math Help - [SOLVED] Find the work done?

1. [SOLVED] Find the work done?

A cylindrical cork, of length $l$ and radius $r$, slowly extracted from the neck of a bottle. If the normal pressure per unit of area between the bottle and the unextracted part of the cork at any instant is constant and is equal to P, show that the work done in extracting it is $\pi \mu rl^2P$, where $\mu$ is the coefficient of friction.

2. A short one for a change!

The solution:

Spoiler:

Let $x$ be the amount of the cork that's been extracted so there is $l-x$ of the length of the cork that is not extracted. This part is subject to a constant pressure of $P$ and this acts on the curved surface of the cork. This surface has area of $2 \pi r (l-x)$ so the total force (normally) against the glass of the bottle is

$F_N = 2 \pi r (l-x) P$

The friction force, $F_{\mu}$, is then

$F_{\mu} = \mu F_N = 2 \mu \pi r (l-x) P$ .

The work, $W$, done is then

$W = \int_{0}^{l} F_{\mu} \, \mathrm{d}x$,

$= 2 \mu \pi r P\int_{0}^{l} (l-x) \, \mathrm{d}x$,

$= 2 \mu \pi r P \left[lx-\frac{x^2}{2}\right]_{x=0}^{x=l}$,

$= \pi \mu r l^2 P$.