Project ForgeAir is contained in a frictionless piston-cylinder arrangement. The atmospheric pressure is 100 kPa and the initial pressure of air in the cylinder is 105 kPa. The area of the piston is 300 cm². Heat is added and the piston moves slowly 8 cm (from its initial position) until it first contacts a linear spring (of stiffness 12.5 N/mm) suspended from a fixed stop above, then compresses that spring a further 8 cm until the piston reaches the stop. Considering the air inside the cylinder as the system, the work interaction is __________ J (round off to the nearest integer).
Explanation
Phase 1 (0 to 8 cm, no spring contact, constant pressure 105 kPa): W1 = pAΔx = 105,000 × 0.03 × 0.08 = 252 J. Phase 2 (8 to 16 cm, spring engaged, force F(x)=P0·A+kx with P0=105 kPa, k=12,500 N/m): W2 = ∫₀^0.08 [P0·A + kx] dx = P0·A(0.08) + k(0.08)²/2 = 252 + 12,500×0.0064/2 = 252+40 = 292 J. Total W = 252+292 = 544 J.
Source: GATE ME 2020 Session 2 (ME-2) (2020)
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