When an external force with angular frequency $\omega_d$ acts on a system of natural angular frequency $\omega$,the system oscillates with angular frequency $\omega_d$. The condition for the amplitude of oscillations to be maximum is

$A$ block of mass $1 \, kg$ attached to a spring is made to oscillate with an initial amplitude of $12 \, cm$. After $2 \, minutes$ the amplitude decreases to $6 \, cm$. Determine the value of the damping constant $b$ for this motion. (Take $\ln 2 = 0.693$)

In forced oscillation of a particle,the amplitude is maximum for a frequency $\omega_{1}$ of the driving force,while the energy is maximum for a frequency $\omega_{2}$ of the driving force. Then:

$A$ body of mass $0.3 \ kg$ hangs by a spring with a force constant of $50 \ N/m$. The amplitude of oscillations is damped and reaches $1/e$ of its original value in about $100$ oscillations. If $\omega$ and $\omega^{\prime}$ are the angular frequencies of undamped and damped oscillations respectively, then the percentage value of $\left(\frac{\omega-\omega^{\prime}}{\omega}\right) \times 100$ is:

Explain the behaviour of the oscillator when the driving frequency is close to the natural frequency in small damped oscillation and define resonance.

$A$ steady force of $120 \ N$ is required to push a boat of mass $700 \ kg$ through water at a constant speed of $1 \ m/s$. If the boat is fastened by a spring and held at $2 \ m$ from the equilibrium position by a force of $450 \ N$,find the angular frequency of damped $SHM$ in $rad/s$.