Question

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

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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+kx−P2A=ma---(i)

If the spring is stretched by a distance x,

V1=A(l2−x)

V2=A(l2+x)

V0=Al2

Since the process is adiabatic,

P0Vγ0=P1Vγ1=P2Vγ2

⇒P1=P0(V0V1)γ=P0(ll−2x)γ

⇒P1=P0(1−2xl)−γ

⇒P1=P0(1+2γxl) ---- (ii)

(using binomial expansion ∵ x<<l)

Similarly P2=P0(1+2xl)−γ

=P0(1−2γxl) --- (iii)

Substituting (ii) and (iii) in (i),

(P1−P2)A+kx=ma

4P0γxAl+kx=ma

⇒x=mak+4P0γAl=4k+4P0γAl

∴n=4

Drawing F.B.D of piston.

P1A+kx−P2A=ma---(i)

If the spring is stretched by a distance x,

V1=A(l2−x)

V2=A(l2+x)

V0=Al2

Since the process is adiabatic,

P0Vγ0=P1Vγ1=P2Vγ2

⇒P1=P0(V0V1)γ=P0(ll−2x)γ

⇒P1=P0(1−2xl)−γ

⇒P1=P0(1+2γxl) ---- (ii)

(using binomial expansion ∵ x<<l)

Similarly P2=P0(1+2xl)−γ

=P0(1−2γxl) --- (iii)

Substituting (ii) and (iii) in (i),

(P1−P2)A+kx=ma

4P0γxAl+kx=ma

⇒x=mak+4P0γAl=4k+4P0γAl

∴n=4

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