Two moles of an ideal monoatomic gas is enclosed in an adiabatic vertical cylinder fitted with a smooth adiabatic massive piston initially at rest. The whole system is placed inside a highly evacuated chamber. If the piston is imparted an initial velocity u=8 m/s in vertical downward direction, after making a large number of oscillations, the piston finally comes to a complete stop. Find the displacement (in meter) of the piston. $({Take g=10 m/s}^{2})$

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answer is 1.28.

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Detailed Solution

Let the vertical upward displacement of the piston before coming to a complete stop is ‘x’
In initial and final equilibrium of the piston
$P A = \frac{n R T_{0}}{1} = \frac{n R T}{(1 + x)} = m g n R T_{0} = m g l ............ (i) n R T_{0} = m g (1 + x) ............ (i i) N o w, n R Δ T + m g x ................ (i i i) \frac{1}{2} m u^{2} = n C_{v} Δ T + m g x \frac{1}{2} m u^{2} = \frac{n 3 R}{2} (T - T_{0}) + m g x \frac{1}{2} m u^{2} = \frac{3 n R Δ T}{2} + m g x \frac{1}{2} m u^{2} = \frac{5}{2} m g x x = \frac{u^{2}}{5 g} = \frac{8 \times 8}{5 \times 10} = 1.28 m e t e r$