The angular frequency of the damped oscillator is given by $\omega = \sqrt{\frac{k}{m} - \frac{r^2}{4m^2}}$,where $k$ is the spring constant,$m$ is the mass of the oscillator,and $r$ is the damping constant. If the ratio $\frac{r^2}{mk}$ is $8\%$,the change in time period compared to the undamped oscillator is approximately as follows:

Which of the diagrams shown in the figure represents the variation of the total mechanical energy of a pendulum oscillating in water as a function of time?

Write the practical examples of resonance.

$A$ 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:

$A$ block of mass $m$ is connected to a light spring of force constant $k$. The system is placed inside a damping medium of damping constant $b$. The instantaneous values of displacement,acceleration and energy of the block are $x, a$ and $E$ respectively. The initial amplitude of oscillation is $A$ and $\omega^{\prime}$ is the angular frequency of oscillations. The incorrect expression related to the damped oscillations is

Write the expression for the mechanical energy of a damped oscillator.