$A$ 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:

Which of the diagrams shown in the figure represents the variation of the total mechanical energy of a pendulum oscillating in water as a function of time?

What are damped oscillations?

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

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 .... $s$

$A$ particle with a restoring force proportional to displacement and a resisting force proportional to velocity is subjected to a driving force $F \sin \omega t$. If the amplitude of the particle is maximum for $\omega = \omega_1$ and the energy of the particle is maximum for $\omega = \omega_2$,then (where $\omega_0$ is the natural frequency of oscillation of the particle):