A monoatomic ideal gas initially at temperature $$\mathrm{T}_1$$ is enclosed in a cylinder fitted with 8 frictionless piston. The gas is allowed to expand adiabatically to a temperature $$\mathrm{T}_2$$ by releasing the piston suddenly. $$\mathrm{L}_1$$ and $$\mathrm{L}_2$$ are the lengths of the gas columns before and after the expansion respectively. Then $$\frac{\mathrm{T}_2}{\mathrm{~T}_1}$$ is

For a monoatomic gas, the work done at constant pressure is '$$\mathrm{W}$$' The heat supplied at constant volume for the same rise in temperature of the gas is

$$[\gamma=\frac{C_p}{C_v}=\frac{5}{2}$$ for monoatomic gas]

A thin metal disc of radius 'r' floats on water surface and bends the surface downwards along the perimeter making an angle '$$\theta$$' with the vertical edge of the dsic. If the weight of water displaced by the disc is '$$\mathrm{W}$$', the weight of the metal disc is [T = surface tension of water]

A particle performing linear S.H.M. of amplitude $$0.1 \mathrm{~m}$$ has displacement $$0.02 \mathrm{~m}$$ and acceleration $$0.5 \mathrm{~m} / \mathrm{s}^2$$. The maximum velocity of the particle in $$\mathrm{m} / \mathrm{s}$$ is