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 the frictional coefficient 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

The correct option is D $\left(4\pi \mu Bb\right)\Delta a$

$P=\frac{N}{A}=\frac{N}{\left(2\pi a\right)b}$

$\Rightarrow$ Stress = $B$ $\times$ strain

$\frac{N}{\left(2\pi a\right)b}=B\frac{2\pi a\Delta a\times b}{\pi a^{2}b}$

$f=\mu N=\mu 4\pi b\Delta aB$