# [SOLVED] Find the work done?

• Jun 15th 2009, 12:03 PM
fardeen_gen
[SOLVED] Find the work done?
A cylindrical cork, of length $\displaystyle l$ and radius $\displaystyle r$, 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 $\displaystyle \pi \mu rl^2P$, where $\displaystyle \mu$ is the coefficient of friction.
• Jun 15th 2009, 03:57 PM
the_doc
A short one for a change!
The solution:

Spoiler:

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

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

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

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

The work, $\displaystyle W$, done is then

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

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

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

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