Question

The figure shows a cylindrical tube of radius $r$ and length $l$ fitted with a cork. The friction coefficient between the cork and the tube is $\mu$. The tube contains an ideal gas at temperature $T$ and atmospheric pressure $P_0$ . It is slowly heated and the cork pops out when the temperature is doubled. What is normal force per unit length exerted by the cork on the periphery of the tube? (Assume uniform temperature throughout the gas and mass of cork to be negligible.)

• A. $\frac{P_{0}A}{2\mu r}$
• B. $\frac{P_{0}A}{\mu \pi r}$
• C. $\frac{P_{0}A}{4\mu \pi r}$
• D. $\frac{P_{0}A}{2\mu \pi r}$

Answer 1

The correct option is D $\frac{P_{0}A}{2\mu \pi r}$

$v\to$ constant
$P\propto T$
As pressure increases ($\uparrow 2P_0$) temperature increases ($\uparrow 2T$)

From the figure (force balancing at the limiting condition)

$2P_{0}A=f_{limiting}+P_{0}A$


from the above figure

$f_{limiting}=P_{0}A=\mu N$ (N is scalar addition of all the normal force)

Normal/Length $=\frac{N}{2\pi r}=\frac{P_{0}A}{\mu 2\pi r}=\frac{P_{0}A}{2\mu \pi r}$