$A$ pendulum with a time period of $1\, s$ is losing energy due to damping. At a certain time,its energy is $45\, J$. If after completing $15$ oscillations,its energy has become $15\, J$,its damping constant (in $s^{-1}$) is:

The amplitude of a mass-spring system,which is executing simple harmonic motion,decreases with time. If mass $m = 500 \, g$,and the decay constant $b = 20 \, g/s$,then how much time $t$ (in seconds) is required for the amplitude of the system to drop to half of its initial value? (Given $\ln 2 = 0.693$)

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

$A$ 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

Why is absolutely pure simple harmonic motion not possible?

$A$ 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 $rad/s$.