Consider a cantilever beam, having negligible mass and uniform flexural rigidity, with length $$0.01m$$. The frequency of vibration of the beam, with a $$0.5$$ kg mass attached at the free tip, is $$100Hz$$. The flexural rigidity (in $$N.{m^2}$$) of the beam is _________.

A rigid uniform rod $$AB$$ of length $$L$$ and mass $$m$$ is hinged at $$C$$ such that $$AC = L/3, CB = 2L/3.$$ Ends $$A$$ and $$B$$ are supported by springs of spring constant $$k.$$ The natural frequency of the system is given by

$$1.5$$ $$kg$$ of water is in saturated liquid state at $$2$$ bar
$$\left( {{v_f} = 0.001061\,\,{m^3}/kg,\,\,{u_f} = 504.0kJ/kg,\,\,{h_f} = 505\,\,kJ/kg} \right).$$ Heat is added in a constant pressure process till the temperature of water reaches
$${400^ \circ }C\left( {v = 1.5493{m^3}/kg,u = 2967.0\,kJ/kg,\,\,h = 3277.0\,kJ/kg} \right).$$ The heat added (in $$kJ$$) in the process is ________.

A reversible heat engine receives $$2$$ $$kJ$$ of heat from a reservoir at $$1000$$ $$K$$ and a certain amount of heat from a reservoir at $$800$$ $$K$$. It rejects $$1$$ $$kJ$$ of heat to a reservoir at $$400$$ $$K.$$ The net work output (in $$kJ$$) of the cycle is