For an under damped harmonic oscillator, resonance

The equation of motion of a harmonic oscillator is given by $${{{d^2}x} \over {d{t^2}}} + 2\xi {\omega _n}{{dx} \over {dt}} + \omega _n^2\,x = 0\,\,\,$$ and the initial conditions at $$t=0$$ are $$\,x\left( 0 \right) = X,{{dx} \over {dt}}\left( 0 \right) = 0.\,\,$$
The amplitude of $$x(t)$$ after $$n$$ complete cycles is

The natural frequency of the system shown below is

Water has a critical specific volume of $$0.003155$$ $${m^3}/kg.$$ A closed and rigid steel tank of volume $$0.025{m^3}$$ contains a mixture of water and steam at $$0.1MPa.$$ The mass of the mixture is $$10$$ $$kg.$$ The tank is now slowly heated. The liquid level inside the tank.