The torque (in N-m) exerted on the crank shaft of a two stroke engine can be described as is the crank angle as $$\,\,T = 10000 + 1000\sin 2\theta - 1200\,\cos 2\theta ,\,\,$$ where $$\,\theta \,\,$$ is the crank angle as measured from inner dead center position. Assuming the resisting torque to be constant, the power (in $$kW$$) developed by the engine at $$100$$ rpm is ________.

Which of the following statements are TRUE for damped vibrations?

$$P.$$ For a system having critical damping, the value of damping ratio is unity and system does not undergo a vibratory motion.
$$Q.$$ Logarithmic decrement method is used to determine the amount of damping in a physical system.
$$R.$$ In case of damping due to dry friction between moving surfaces resisting force of constant magnitude acts opposite to the relative motion.
$$S.$$ For the case of viscous damping, drag force is directly proportional to the square of relative velocity.

Figure shows a single degree of freedom system. The system consists of a mass less rigid bar $$OP$$ hinged at $$O$$ and a mass $$m$$ at end $$P.$$ The natural frequency of vibration of the system is

A rigid container of volume $$0.5$$ $${m^3}$$ contains $$1.0$$ $$kg$$ of water at $${120^ \circ }C$$
$$\left( {{V_f} = 0.00106\,{m^3}/kg,\,\,{v_g} = 0.8908\,\,{m^3}/kg} \right).$$ The state of water is