Consider the system in the figure below. Initially the gas in the cylinder is at high pressure P_i and the piston held in place at volume V_i with a lock. At time zero the lock is released and the gas allowed to expand, accelerating the piston of mass M_p to the left. Friction is negligible. The expansion may be idealized as an adiabatic polytropic process with ∝ = 1.4, and there is negligible leakage. The pressure on the left side of the piston may be taken as a constant P₀ during the process.

(a) Derive an expression for the velocity of the piston Ѵ_p as a function of the instantaneous gas volume V; the symbols defined above (and no others) will appear as parameters in your analysis.(b) Plot Ѵ_p versus V and P versus V for the “can-thrower” case M_p = 0.5 kg, P₀ = 105 Pa, P_i = 7 × 10⁵ Pa, V_i = 10⁻³ m³, covering the range for which P ≥ P₀.(c) Specify the cylinder length required to produce 40 m/s when P = P₀ for the canthrower, assuming the cylinder diameter is 8 cm.

Transcribed Image Text:

Po Mp Pi latch