Question

A bottle has an opening of radius $a$ and length $b$. A cork of length $b$ and radius $(a+\Delta a)$ where $(\Delta a<<a)$ is compressed to fit into the opening completely (See figure). If the bulk modulus of cork is $B$ and frictional coefficient between the bottle and cork is $\mu$ then the force needed to push the cork into the bottle is :

• A. $(2\pi \mu Bb)\Delta a$
• B. $(\pi \mu Bb)\Delta a$
• C. $(\pi \mu Bb)a$
• D. $(4\pi \mu Bb)\Delta a$

Answer 1

Answer: (D)

$\beta \frac{\Delta V}{V}=-\Delta P$
$V_i=\pi (a+\Delta a{)}^{2}b$
$V_f=\pi a^{2}b$
$\Delta V\simeq -2\pi ab\Delta a$
$\frac{\Delta V}{V}=\frac{-2\pi ab\Delta a}{\pi a^{2}b}=\frac{-2\Delta a}{a}$
$\Rightarrow \Delta P=\frac{2\beta \Delta a}{a}$
Normal force $=\frac{2\beta \Delta a}{a}2\pi \alpha b$
$=4\pi \beta b\Delta a$
friction $=\mu N$
$=4\pi \mu \beta b\Delta a$