Damped Oscillations

A particle is oscillating freely with a natural frequency ${\omega }_0$ and amplitude $a$. It is later subjected to a damping force proportional to its velocity and keeps oscillating with a frequency $\omega .$ Which of the following statement is true?

A simple pendulum is set into vibrations. The bob of the pendulum comes to rest after some time due to 

The amplitude of a damped oscillator decreases to $0.9$ times its original magnitude is $5 \mathrm{s}$. In another $10 \mathrm{s}$ it will decrease to $\mathrm{\alpha }$ times its original magnitude, where $\alpha$ equals. 

If a simple pendulum has significant amplitude (up to a factor of $\frac{1}{e}$ of original) only in the period between $t=0 \mathrm{s}$ to $t=\tau \mathrm{s}$ then $\tau$ may be called the average life of the pendulum. When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity, with $b$ as the constant of proportionality, the average lifetime of the pendulum is (assuming damping is small) in seconds

Which of the following differential equations represents a damped harmonic oscillator?

When a damped harmonic oscillator completes $100$ oscillations, its amplitude is reduced to ${\left(\frac{1}{3}\right)}^{\mathrm{rd} }$ of its initial value. What will be its amplitude when it completes $200$ oscillations?
 

A damped oscillator consists of a spring-mass system with mass $2 \mathrm{kg}$ and spring of spring constant $10 \mathrm{N} {\mathrm{m}}^{-1}$. The damping force is given by $F=-b\left(\frac{\mathrm{d}x}{dt}\right)$ where $b=280 \mathrm{g} {\mathrm{s}}^{-1}.$ The time required for the amplitude of the oscillations to reduce to one-fourth ${\left(\frac{1}{4}\right)}^{th}$ of its initial value is: (Assume $\ln 2=0.7$)

The amplitude of a damped oscillator becomes one third in $2 \sec .$ If its amplitude after $6 \sec$ is $1/n$ times the original amplitude then the value of $n$ is

An oscillator of mass $10 \mathrm{g}$ is oscillating with natural frequency of 100 Hz. Under slight damped conditions, a periodic force, $F=100\cos 20\pi t$ is applied on it. The amplitude of oscillation is approximately,

The amplitude of a S.H.M. reduces to $\frac{1}{3}$ in first $20 \mathrm{s}$. Then in first $40 \mathrm{s}$ its amplitude becomes -

When an oscillator completes $100$ oscillation its amplitude is reduced to $\frac{1}{3}$ of initial value. What will be its amplitude, when it completes $200$ oscillation: - 

Amplitude of a damped oscillator reduces to$0.9$ times its original magnitude in $5 \mathrm{s}$. In another $10 \mathrm{s}$, it decreases to $\alpha$ times to its original magnitude. Find the value of $\alpha$.

A block of mass $200\mathrm{ }\mathrm{g}$ is executing $\mathrm{S}\mathrm{H}\mathrm{M}$ under the influence of a spring of spring constant $\mathrm{k}=90\mathrm{ }\mathrm{N} {\mathrm{m}}^{-1}$ and a damping constant $\mathrm{b}=40\mathrm{ }\mathrm{g} {\mathrm{s}}^{-1}$. The time elapsed for its amplitude to drop to half of its initial value is (Given, $\ln \left(\frac{1}{2}\right)=-0.693$)

In the following a statement of Assertion is followed by a statement of Reason.

Assertion: In damped oscillations, the oscillator experiences both conservative and non-conservative forces.

Reason: In damped oscillations mechanical energy of oscillator decreases with time.

In an experiment to find the loss of energy with respect to time in the case of a swinging simple pendulum, the graph between the square of amplitude and time is best represented by

To study the dissipation of energy student Plots a graph between square root of time and amplitude. The graph would be a -

The amplitude of damped oscillator becomes $\frac{1}{2}$ in $2 \mathrm{s}$. Its amplitude after $6$ is $\frac{1}{n}$ times the original. Then $n$ is equal to

The amplitude of damped oscillator becomes $\frac{1}{3}\mathrm{rd}$ of the original in $2 \mathrm{s}$. Its amplitude after $6 \mathrm{s}$ is $\frac{1}{n}$ times the original. Then, $n$ is equal to,

The amplitude of a damped oscillator becomes ${\left(\frac{1}{3}\right)}^{\mathrm{rd}}$ in $2 \mathrm{s}$. If its amplitude after $6 \mathrm{s}$ is $\frac{1}{n}$ times the original amplitude, the value of $n$ is