A uniform stiff rod of length $$300$$ $$mm$$ and having a weight of $$300$$ $$N$$ is pivoted at one end and connected to a spring at the other end. For keeping the rod vertical in a stable position the minimum value of spring constant $$K$$ needed is

A mass M, of 20 kg is attached to the free end of a steel cantilever beam of length 1000 mm having a cross-section of 25 $$\, \times \,$$ 25 mm. Assume the mass of the cantilever to be negligible and E_steel = 200 Gpa. If the lateral vibration of this system is critically damped using a viscous damper, the damping constant of the damper is

A gas contained in a cylinder is compressed, the work required for compression being $$5000$$ $$kJ.$$ During the process, heat interaction of $$200$$ $$kJ$$ causes the surroundings to the heated. The change in internal energy of the gas during the process is

A heat engine having an efficiency of $$70\% $$ is used to drive a refrigerator having a coefficient of performance of $$5.$$ The energy absorbed from low temperature reservoir by the refrigerator for each $$kJ$$ of energy absorbed from high temperature source by the engine is