A uniform cylinder of length $$L$$ and mass $$M$$ having cross-sectional area $$A$$ is suspended, with its length vertical, from a fixed point by a mass-less spring such that it is half submerged in a liquid of density $$\sigma $$ at equilibrium position. The extension $${x_0}$$ of the spring when it is in equilibrium is:

Assume that a drop of liquid evaporates by decreases in its surface energy, so that its temperature remains unchanged. What should be the minimum radius of the drop for this to be possible ? The surface tension is $$T,$$ density of liquid is $$\rho $$ and $$L$$ is its latent heat of vaporization.

The above $$p$$-$$v$$ diagram represents the thermodynamic cycle of an engine, operating with an ideal monatomic gas. The amount of heat, extracted from the source in a single cycle is

If a piece of metal is heated to temperature $$\theta $$ and then allowed to cool in a room which is at temperature $${\theta _0},$$ the graph between the temperature $$T$$ of the metal and time $$t$$ will be closest to