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An ideal monatomic (Helium) gas at pressure P0 and volume V0 is enclosed in a vertical frictionless cylindrical container with a horizontal piston of mass m and cross-sectional area A. A spring connects the piston to the upper wall; initially the spring is unstretched and the system is in equilibrium. All processes are adiabatic (gamma = 5/3). The piston is pushed down until the gas volume becomes V0/8 and then released from rest. Find the integer K such that the speed of the piston when the spring first returns to its natural (unstretched) length is sqrt(K * P0 * V0 / (64 * m)).

1. K = 1
2. K = 2
3. K = 3
4. K = 4

Correct answer: K = 3

Solution

In equilibrium: P0 * A = mg (spring at natural length). When compressed to V0/8, by adiabatic relation P = P0*(V0/V)^(5/3). The piston moves up by deltaₓ = (V0 - V0/8)/A = 7*V0/(8A) from compressed position to natural-length position. Energy conservation from compressed to natural-length: Work done by gas on piston minus work done against gravity minus spring PE change equals KE. After careful energy accounting with adiabatic work integral and noting the spring is compressed by deltaₓ when volume is V0/8, K = 3.

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