Physics

Oscillations and Simple Harmonic Motion

138 Questions

Oscillations and simple harmonic motion focus on amplitude, damped vibrations, and force equations. These physics principles are essential for various engineering and civil services examinations. Review these problems to understand the core mechanics of oscillating bodies.

Damped harmonic oscillationSHM amplitudeForce equationsForced oscillationVibratory motion concepts

Oscillations and Simple Harmonic Motion Questions

Multiple choice physics rigid body dynamics motion of rigid body rigid body equilibrium of a rigid body

For a particle showing motion under the force $F=-5{ \left( x-2 \right)  }^{ 2 },$ the motion is

  1. Translatory

  2. Oscillatory

  3. SHM

  4. All of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The force is F = -5(x-2)^2. For motion to be oscillatory, there must be a stable equilibrium point and a restoring force. Since the force is proportional to the square of the displacement, it is not SHM (which requires F proportional to x), but it is oscillatory as it acts to restore the particle toward the equilibrium position x=2.

Multiple choice physics rigid body dynamics motion of rigid body rigid body equilibrium of a rigid body

For a particle showing motion under the force $F=-5{ \left( x-2 \right) },$ the motion is

  1. Translatory

  2. Oscillatory

  3. SHM

  4. Both (2) & (3)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The force F = -5(x-2) is a linear restoring force of the form F = -k(x-x0). This is the definition of Simple Harmonic Motion (SHM). Since all SHM is also oscillatory, both options (2) and (3) are correct.

Multiple choice physics free, damped and forced oscillations resonance: examples and uses resonance oscillatory motion

Resonance is a special case of $\underline{           }$ vibrations, when frequency of the driving force is$\underline{           }$ natural frequency of the body.

  1. forced, equal to the

  2. particle , more then the

  3. wave, equal to the

  4. forced, less than the
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Resonance is a special case of forced vibrations, when frequency of the driving force is equal to the natural frequency of the body.
Resonance is the result of an external force vibrating at the same frequency as the natural frequency of a system. Natural frequency is a characteristic of every machine, structure and even animals. Often, resonance can be confused with the natural frequency or critical frequency. If equipment is operating in a state of resonance, the vibration levels will be amplified significantly, which can cause equipment failure and plant downtime. It is, therefore, important that the running speed of equipment be out of the resonance range.

Multiple choice physics free, damped and forced oscillations resonance: examples and uses resonance oscillatory motion

A forced oscillator is acted upon by a force $F={ F } _{ \circ  }sin\omega t.$. The amplitude of oscillation is given by $\displaystyle\frac{55}{\sqrt{2\omega^2 - 36\omega + 9}}$. The resonant angular frequency is

  1. $2\space unit$
  2. $9\space unit$
  3. $18\space unit$
  4. $36\space unit$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

At resonance, amplitude of oscillation is maximum
$\quad \Rightarrow 2\omega^2 - 36\omega + 9$ is maximum
$\quad \Rightarrow 4\omega - 36 = 0$ (derivative is zero)
$\quad \Rightarrow \omega = 9$

Multiple choice physics superposition of waves-2: stationary (standing) waves: vibrations of air columns from moving to stationary stationary (or standing) waves formation of stationary waves

List - I                                                        List - II
a)  Phase difference                                 e) $\pi $
between two particles in
 alternate loops.
b)  Phase difference                                 f)  $\displaystyle \frac{\pi}{2}$
 between two particles in
successive loops
c)  Phase difference between                g) $2\pi $
two particles in the same loop
d)  Phase difference between                h) $0$
$Y _{1}=a\sin (\omega t-Kx)$
$Y _{2}=a\cos (\omega t-Kx)$

  1. a-g, b-e, c-h, d-f

  2. a-e, c-f, d-g, e-h

  3. a-f, b-e, c-g, d-h

  4. a-g, b-e, c-f, d-h
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

a ) Two particles in alternate loops refers to particles having same phase and direction
$\therefore $ phase difference is $2\pi $
b ) Two particles in successive loops differs in phase by $\pi $ 
c ) Two particles in same loop are always in phase $\Rightarrow $ phase difference is $0$.
d ) $y _1= a  \sin  (\omega t-kx)$
$y _2= a  \cos  (\omega t-kx)$
$= a  \sin (\omega t-kx +\dfrac{\pi}{2})$
$\Rightarrow $ phase difference is $ \dfrac{\pi }{2}$.

Multiple choice free, damped and forced oscillations free, forced and damped oscillations oscillations oscillation and waves physics

$mx^{2} - bx + k = 0$. Find time after which to the energy will become half of initial maximum value in damped forced oscillation.

  1. $t = \dfrac {m}{b} + \dfrac {1}{2} ln2$
  2. $t = \dfrac {m}{b} \times \dfrac {2}{3} ln2$
  3. $t = \dfrac {m}{b} - \dfrac {1}{2} ln2$
  4. $t = \dfrac {m}{b} \times \dfrac {1}{2} ln2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\dfrac {1}{\sqrt {2}} = e^{-bt/m}$
$ln \sqrt {2} = \dfrac {bt}{m}$
$t = \dfrac {m}{b} \times \dfrac {1}{2} ln2$.

Multiple choice free, damped and forced oscillations free, forced and damped oscillations oscillations oscillation and waves physics

A particle of mass 0.10 kg executes Simple harmonic motion with an amplitude 0.05 m and frequency 20 vib/s. Its energy of oscillation is

  1. 2 J

  2. 4 J

  3. 1 J

  4. zero
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$E = \frac{1}{2}m{\omega^2}{A^2}$

$ = \frac{1}{2} \times 10 \times {\left( {2\pi  \times 20} \right)^2} \times {\left( {0.05} \right)^2}$
$ = 2J$
Hence,
option $(A)$ is correct answer.

Multiple choice free, damped and forced oscillations free, forced and damped oscillations oscillations oscillation and waves physics

Forced oscillation is 

  1. simple harmonic motion but driven externally

  2. simple harmonic motion without driven externally

  3. having resonance when the driving frequency is the same as the natural frequency of the swing.

  4. Both A and C
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The force oscillation is (1) a simple harmonic motion driven by an external agency 

(ii) It has a resonance when the natural frequency of motion is equal to the driven frequency. 
At resonance the amplitude of motion and velocity of motion become maximum.

Multiple choice free, damped and forced oscillations free, forced and damped oscillations oscillations oscillation and waves physics

A driven oscillator is acted upon by a force $F={ F } _{ 0 }sin\ \omega $. The amplitude of oscillation is given by $A=\frac { { F } _{ 0 } }{ \sqrt { a{ \omega  }^{ 2\  }  -b\omega \ +c }} $, the resonant angular frequency is

  1. $d\frac { a }{ b }$
  2. $\dfrac { 2a }{ b } $
  3. $\dfrac { b }{a } $
  4. $\dfrac { b }{ 2a } $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The driven force $ F= {F} _{0} \sin \omega t$

The amplitude of oscillation 
$A=\dfrac { { F } _{ 0 } }{ \sqrt { { a\omega  }^{ 2 }-b\omega +c }  } $
At resonance frequency A become maximum.So,
$\dfrac { dA }{ d\omega  } =0\ \Rightarrow { F } _{ 0 }\left( \dfrac { 2a\omega -b }{ \left( a{ \omega  }^{ 2 }-b\omega +1 \right)^{ \dfrac { 3 }{ 2 }}  }  \right) =0\ \Rightarrow \omega =\dfrac{b}{2a} \rightarrow$  
This is the resonence frequency.

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

The amplitude of a damped harmonic oscillator becomes halved in $\ minute$. After three minutes, the amplitude will becomes $\dfrac{1}{x}$ of initial amplitude, where $x$ is ?

  1. $8$
  2. $2$
  3. $3$
  4. $4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a damped oscillator, amplitude A(t) = A0 * exp(-bt/2m). Given A(1) = A0/2, then exp(-b/2m) = 1/2. After 3 minutes, A(3) = A0 * (exp(-b/2m))^3 = A0 * (1/2)^3 = A0/8. Thus x = 8.

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

A particle performing SHM is found at its equilibrium at $  t=1\ sec$ and it is found to have a speed of $0.25 \mathrm{m} / \mathrm{s}  $ at $  \mathrm{t}=2\ \mathrm{sec}  $ . If the period of oscillation is $6\ \mathrm{sec}  $. Calculate amplitude of oscillation

  1. $ \frac{3}{2 \pi} \mathrm{m} $
  2. $ \frac{3}{ \pi} \mathrm{m} $
  3. $ \frac{6}{2 \pi} \mathrm{m} $
  4. $ \frac{6}{ \pi} \mathrm{m} $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In SHM, x(t) = A sin(omega * t + phi). Equilibrium at t=1 means sin(omega + phi) = 0. Period T=6s, so omega = 2pi/6 = pi/3. At t=2, v = A * omega * cos(omega * t + phi) = 0.25. Solving these equations yields A = 3/(2pi).