$A$ 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:

What is pure simple harmonic oscillation? Why is it not $100 \%$ possible in practice?

$A$ particle of mass $m$ is attached to a spring (of spring constant $k$) and has a natural angular frequency $\omega_0$. An external force $F(t)$ proportional to $\cos \omega t$ (where $\omega \neq \omega_0$) is applied to the oscillator. The displacement of the oscillator will be proportional to:

The amplitude of a damped oscillator becomes half in $1$ minute. The amplitude after $3$ minutes will be $\frac{1}{x}$ times the original. Then $x$ is

The displacement of a damped harmonic oscillator is given by $x(t) = e^{-0.1t} \cos(10\pi t + \varphi)$. The time taken for its amplitude of vibration to drop to half of its initial value is close to .... $s$

When an external force with angular frequency $\omega_d$ acts on a system of natural angular frequency $\omega$,the system oscillates with angular frequency $\omega_d$. The condition for the amplitude of oscillations to be maximum is