If the amplitude of a damped harmonic oscillator becomes half of its initial amplitude in a time of $10 \ s$, then the time taken for the mechanical energy of the oscillator to become half of its initial mechanical energy is (in $s$)

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

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

$Assertion :$ Resonance is a special case of forced vibration in which the natural frequency of vibration of the body is the same as the impressed frequency of external periodic force and the amplitude of forced vibration is maximum.
$Reason :$ The amplitude of forced vibrations of a body increases with an increase in the frequency of the externally impressed periodic force.

For a damped harmonic oscillator governed by the equation $m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0$, find the time $t$ after which the mechanical energy becomes half of its initial maximum value.

$A$ 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