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

Answer 1

Answer: (D)

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