Question

An adiabatic piston of mass $m$ equally divides a diathermic container of volume $V_0$ and length $l$. A light spring connects the piston to the right wall. At equilibrium, pressure on each side of the piston is $P_0$. The container starts moving with acceleration $a$ towards the right. Then,

Assume that $x<<l$, the gas in the container has the adiabatic exponent (ratio of $C_P$ and $C_V)\gamma ,m=2\text{kg},a=2{\text{m/s}}^2$

• A. The piston will move with acceleration $a$ at equilibrium.
• B. The stretch $x$ of the spring when the acceleration of the piston equals the acceleration of the container is $\frac{ma}{k+\frac{2P_0\gamma A}{l}}$.
• C. The difference in pressure between the two compartments is $\frac{4P_0\gamma A}{l}$
• D. The stretch $x$ of the spring when the acceleration of the piston equals the acceleration of the container is $\frac{ma}{k+\frac{4P_0\gamma A}{l}}$.
  

Answer 1

Answer: (A), (C), (D)

The correct options are
A The piston will move with acceleration $a$ at equilibrium.
C The difference in pressure between the two compartments is $\frac{4P_0\gamma A}{l}$
D The stretch $x$ of the spring when the acceleration of the piston equals the acceleration of the container is $\frac{ma}{k+\frac{4P_0\gamma A}{l}}$.


Drawing F.B.D of piston.

$P_{1}A+kx-P_{2}A=ma$---(i)


If the spring is stretched by a distance $x$,

$V_1=A(\frac{l}{2}-x)$
$V_2=A(\frac{l}{2}+x)$
$V_0=A\frac{l}{2}$

Since the process is adiabatic,

$P_0{V}_0^{\gamma }=P_1{V}_1^{\gamma }=P_2{V}_2^{\gamma }$
$\Rightarrow P_1=P_0{\left(\frac{V_0}{V_1}\right)}^{\gamma }=P_0{\left(\frac{l}{l-2x}\right)}^{\gamma }$
$\Rightarrow P_1=P_0{(1-\frac{2x}{l})}^{-\gamma }$
$\Rightarrow P_1=P_0(1+\frac{2\gamma x}{l})$ ---- (ii)
(using binomial expansion $\because x<<l)$

Similarly $P_2=P_0{(1+\frac{2x}{l})}^{-\gamma }$
$=P_0(1-\frac{2\gamma x}{l})$ --- (iii)

Substituting (ii) and (iii) in (i),
$(P_1-P_2)A+kx=ma$
$\frac{4P_0\gamma xA}{l}+kx=ma$
$\Rightarrow x=\frac{ma}{k+\frac{4P_0\gamma A}{l}}=\frac{4}{k+\frac{4P_0\gamma A}{l}}$
$\therefore n=4$