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

A wooden wheel of radius $$R$$ is made of two semicircular part (see figure). The two parts are held together by a ring made of a metal strip of cross sectional area $$S$$ and length $$L.$$ $$L$$ is slightly less than $$2\pi R.$$ To fit the ring on the wheel, it is heated so that its temperature rises by $$\Delta T$$ and it just steps over the wheel. As it cools down to surrounding temperature, it process the semicircular parts together. If the coefficient of linear expansion of the metal is $$\alpha $$, and its Young's modulus is $$Y,$$ the force that one part of the wheel applies on the other part is :

Helium gas goes through a cycle $$ABCD$$ (consisting of two isochoric and isobaric lines) as shown in figure efficiency of this cycle is nearly : (Assume the gas to be close to ideal gas)

A Carnot engine, whose efficiency is $$40\% $$, takes in heat from a source maintained at a temperature of $$500$$ $$K.$$ It is desired to have an engine of efficiency $$60\% .$$ Then, the intake temperature for the same exhaust (sink) temperature must be :