In case of damped oscillation,the frequency of oscillation is ............

The time taken for the amplitude of vibrations of a damped oscillator to drop to $25 \%$ of its initial value is $t$. The time taken for its mechanical energy to drop to $50 \%$ of its initial mechanical energy is

For the damped oscillator shown in the figure,the mass $m$ of the block is $200 \; g$,$k = 90 \; N m^{-1}$,and the damping constant $b$ is $40 \; g s^{-1}$. Calculate:
$(a)$ the period of oscillation,
$(b)$ the time taken for its amplitude of vibrations to drop to half of its initial value,and
$(c)$ the time taken for its mechanical energy to drop to half its initial value.

$A$ simple pendulum oscillates in air with time period $T$ and amplitude $A$. As the time passes,

Derive the differential equation for damped oscillations and write its solution.

$A$ simple pendulum is set into vibrations. The bob of the pendulum comes to rest after some time due to