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An adiabatic piston of mass m equally divides a diathermic container of volume V0 and length l. A light spring connects the piston to the right wall. At equilibrium, pressure on each side of the piston is P0. 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 CP and CV)γ, m=2 kg, a=2 m/s2


A
The piston will move with acceleration a at equilibrium.
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B
The stretch x of the spring when the acceleration of the piston equals the acceleration of the container is mak+2P0γAl.
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C
The difference in pressure between the two compartments is 4P0γAl
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D
The stretch x of the spring when the acceleration of the piston equals the acceleration of the container is mak+4P0γAl.
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Solution

The correct options are
A The piston will move with acceleration a at equilibrium.
C The difference in pressure between the two compartments is 4P0γAl
D The stretch x of the spring when the acceleration of the piston equals the acceleration of the container is mak+4P0γAl.

Drawing F.B.D of piston.

P1A+kxP2A=ma---(i)


If the spring is stretched by a distance x,

V1=A(l2x)
V2=A(l2+x)
V0=Al2

Since the process is adiabatic,

P0Vγ0=P1Vγ1=P2Vγ2
P1=P0(V0V1)γ=P0(ll2x)γ
P1=P0(12xl)γ
P1=P0(1+2γxl) ---- (ii)
(using binomial expansion x<<l)

Similarly P2=P0(1+2xl)γ
=P0(12γxl) --- (iii)

Substituting (ii) and (iii) in (i),
(P1P2)A+kx=ma
4P0γxAl+kx=ma
x=mak+4P0γAl=4k+4P0γAl
n=4

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