Tag: oscillations

Questions Related to oscillations

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass M. The piston and the cylinder have equal cross sectional area A. When the piston is in equilibrium, the volume of the gas $ \mathrm{V} _{0}  $ and its pressure is $  \mathrm{P} _{0} $ The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency.

physics oscillations introduction to sound free, forced and damped oscillations resonance

  1. $ \dfrac{1}{2 \pi} \dfrac{\mathrm{A} \gamma P _{0}}{V _{0} M} $

  2. $ \dfrac{1}{2 \pi} \dfrac{V _{0} M P _{0}}{A^{2} \gamma} $

  3. $ \dfrac{1}{2 \pi} \sqrt{\dfrac{A^{2} \gamma P _{0}}{M V _{0}}} $

  4. $ \dfrac{1}{2 \pi} \sqrt{\dfrac{M V _{0}}{A \gamma P _{0}}} $

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Correct Option: C

The amplitude of a damped oscillator becomes half on one minute. The amplitude after 3 minute will be $\displaystyle\dfrac{1}{X}$ times the original, where $X$ is

physics oscillations introduction to sound free, forced and damped oscillations resonance

  1. $2\times 3$

  2. $2^3$

  3. $3^2$

  4. $3\times 2^2$

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Correct Option: B

The equation of a damped simple harmonic motion is $ m \frac {d^2x}{dt^2} + b \frac {dx}{dt} + kx=0 . $ Then the angular frequency of oscillation is:

physics oscillations introduction to sound free, forced and damped oscillations resonance

  1. $ \omega = ( \frac {k}{m}+\frac {b}{4m})^{1/2} $

  2. $ \omega = ( \frac {k}{m}-\frac {b}{4m})^{1/2} $

  3. $ \omega = ( \frac {k}{m}+\frac {b^2}{4m})^{1/2} $

  4. $ \omega = ( \frac {k}{m}-\frac {b^2}{4m^2})^{1/2} $

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Correct Option: D

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

physics oscillations introduction to sound free, forced and damped oscillations resonance

  1. $0.7$

  2. $0.81$

  3. $0.729$

  4. $0.6$

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Correct Option: C

A lightly damped oscillator with a frequency $\left( \omega  \right) $  is set in motion by harmonic driving force of frequency $\left( n \right) $. When $n\ll \omega $, then response of the oscillator is controlled by

physics oscillations introduction to sound free, forced and damped oscillations resonance

  1. Oscillator frequency

  2. spring constant

  3. Damping coefficient

  4. Inertia of the mass

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Correct Option: B,C

On account of damping , the frequency of a vibrating body

physics oscillations introduction to sound free, forced and damped oscillations resonance

  1. remains unaffceted

  2. increases

  3. decreases

  4. changes erratically

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Correct Option: C
Explanation:

Damping is caused by opposing force, which decreases the frequency.

Ans: C

In damped oscillations, the amplitude after $50$ oscillations is $0.8\;a _0$, where $a _0$ is the initial amplitude, then the amplitude after $150$ oscillations is

physics oscillations introduction to sound free, forced and damped oscillations resonance

  1. $0.512\;a _0$

  2. $0.280\;a _0$

  3. Zero

  4. $a _0$

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Correct Option: A
Explanation:

The amplitude, a, at time $t$ is given by $a=a _0\;exp(-\,\alpha t)$



$a _{50}=a _0\;exp(-\alpha\times 50T)=0.80\;a _0$



where $T$ is the period of oscillation



$a _{150}=a _0\;exp(-a\times 150T)$



$=a _0\;(0.8)^3=0.512\,a _0$

When an oscillator completes $100$ oscillations its amplitude reduces to $\displaystyle\dfrac{1}{3}$ of its initial value. What will be its amplitude when it completes $200$ oscillations?

physics oscillations introduction to sound free, forced and damped oscillations resonance

  1. $\displaystyle\dfrac{1}{8}$

  2. $\displaystyle\dfrac{2}{3}$

  3. $\displaystyle\dfrac{1}{6}$

  4. $\displaystyle\dfrac{1}{9}$

Show answer
Correct Option: D
Explanation:


Its is a damped oscillation, where amplitude of oscillation at time $t$ is given by $A = a _0e^{-\gamma t}$
where $a _0 = $ initial amplitude of oscillation
$\quad \gamma = $ damping constant
As per question, $\displaystyle\dfrac{a _0}{3} = a _0e^{-\gamma100/v}\quad                    ...(i)$
(where $v$ is the frequency of oscillation)
and $A = a _0e^{-\gamma200/v} \quad                ...(ii)$
From $(i)$; $\quad \displaystyle\dfrac{a _0}{3} = a _0e^{-\gamma\times100/v} \quad            ...(iii)$
Dividing equation $(ii)$ by $(iii)$, we have

$\quad \displaystyle\dfrac{A}{a _0(1/3)} = \displaystyle\dfrac{e^{-\gamma\times200/v}}{e^{-\gamma\times100/v}} = e^{-\gamma\times100/v} = \displaystyle\dfrac{1}{3}$

or $A = a _0\times\displaystyle\dfrac{1}{3}\times\displaystyle\dfrac{1}{3} = \displaystyle\dfrac{1}{9}a _0$


Two point masses $m _1$ and $m _2$ are coupled by a spring of spring. Constant $k$ and uncompressed length $L _0$. The spring is fully compressed and a thread ties the masses together with negligible separation between them. The tied assembly is moving in the $+x$ direction with uniform speed $v _0$. At a time, say $t = 0$, it is passing the origin and at that instant the thread breaks. The masses, attached to the spring, start oscillating. The displacement of mass $m _1$ given by $x _1(t) = v _0 t(1 - cos \omega t)$ where $A$ is a constant. Find (i) the displacement $x _2(t)$ is $m _2$, and (ii) the relationship between $A$ and $L _0$.

physics oscillations introduction to sound free, forced and damped oscillations resonance

  1. (i) $v _0 t + \dfrac{m _1}{2m _2}A(1 - cos \omega t)$

    (ii) $A = \left(\dfrac{m _2}{2m _1 + m _2}\right)$

  2. (i) $v _0 t + \dfrac{m _1}{m _2}A(1 - cos \omega t)$

    (ii) $A = \left(\dfrac{m _2}{m _1 + m _2}\right)$

  3. (i) $v _0 t + \dfrac{m _1}{3m _2}A(1 - cos \omega t)$

    (ii) $A = \left(\dfrac{m _2}{3m _1 + m _2}\right)$

  4. (i) $v _0 t + \dfrac{m _1}{4m _2}A(1 - cos \omega t)$

    (ii) $A = \left(\dfrac{m _2}{4m _1 + m _2}\right)$

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Correct Option: B

The natural angular frequency of a particle of mass 'm' attached to an ideal spring of force constant 'K' is

horizontal oscillations of a mass attached to a spring oscillations due to a spring simple harmonic motion oscillations physics

  1. $\sqrt{\frac{K}{m}}$

  2. $\sqrt{\frac{m}{K}}$

  3. $\left ( \frac{K}{m} \right )^{2}$

  4. $\left ( \frac{m}{K} \right )^{2}$

Show answer
Correct Option: A
Explanation:
Suppose you displace the particle by a distance $'x'$
The spring now exerts a force,
This provides nccenary force for $SHM$
$\Rightarrow \ F=mwe^2x=k2$ ($w:$ natural angular frequency )
$\Rightarrow \ w=\sqrt {K/m}$