The amplitude of vibration of a particle is given by $a_m = \frac{a_0}{a\omega^2 - b\omega + c}$,where $a_0, a, b,$ and $c$ are positive constants. The condition for a single resonant frequency is:

$A$ block of mass $100 \,g$ is connected to an elastic spring of spring constant $450 \,N m^{-1}$ and oscillates vertically. The block-spring system is in a viscous surrounding medium with a damping constant $b = 69.3 \,g \,s^{-1}$. Calculate the time in which the amplitude of oscillations drops to half of its initial value. (Take $\ln 2 = 0.693$) (in $\,s$)

In forced oscillations,a particle oscillates simple harmonically with a frequency equal to

$Assertion :$ The amplitude of an oscillating pendulum decreases gradually with time.
$Reason :$ The frequency of the pendulum decreases with time.

$Assertion :$ 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.

Which of the following figures represents damped harmonic motion?