Question

A bottle has an opening of radius $a$ and length $b$ . A cork of length $b$ and radius $\left(a+\Delta a\right)$ where $\left(\Delta a\ll a\right)$ , is compressed to fit into the opening completely (see figure). If the bulk modulus of cork is $B$ and the coefficient of friction between the bottle and cork is $\mu$ , then the force needed to push the cork into the bottle is

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

Answer 1

Answer: (D)

$Pressure=\frac{N}{A}=\frac{N}{\left(2\pi a\right)b}$
$\Rightarrow stress=B\times strain$
$\frac{N}{\left(2\pi a\right)b}=B\frac{2\pi a\Delta a\times b}{{\left(\pi a\right)}^{2}b}$
$\Rightarrow N=B$
$f=\mu N=\mu 4\pi bB\Delta a$