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. $(2 \pi \mu B\, b) \Delta\,a$
• B. $(\pi \mu B\, b) \Delta a$
• C. $(\pi \mu \,B\, b ) a$
• D. $(4 \pi \mu\, B \,b ) \Delta \,a$

Answer 1

Answer: (D)

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