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  4. Free, Forced And Damped Oscillation - Practice Questions & MCQ

Free, Forced And Damped Oscillation - Practice Questions & MCQ

Updated on Sep 18, 2023 18:34 AM

Quick Facts

  • Damped Harmonic motion is considered one the most difficult concept.

  • 14 Questions around this concept.

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QuestionMEDIUM

 A pendulum with time period of 1s is losing energy due to damping. At 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 :

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The amplitude of a simple pendulum,  oscillating in air with a small spherical bob,   decreases from 10 cm to 8 cm in 40 seconds.Assuming that Stokes law is valid, and ratio of the coefficient of viscosity of air to that of carbon dioxide is 1.3, the time in which amplitude of this pendulum will  reduce  from  10  cm  to  5  cm  in  carbondioxide will be close to\left ( ln\: 5 = 1.601,ln\: 2= 0.693 \right )    

 

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

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Concepts Covered - 0

Damped Harmonic motion

Free\undamped oscillation-

  • The oscillation of a particle with fundamental frequency under the influence of restoring force is defined as free oscillations.
  • The amplitude, frequency, and energy of oscillation remain constant.
  • The frequency of free oscillation is called natural frequency because it depends upon the nature and structure of the body.

Damped oscillation-

  • The oscillation of a body whose amplitude goes on decreasing with time is defined as damped oscillation.
  • The amplitude of these oscillations decreases exponentially (as shown in the below figure) due to damping forces like frictional force, viscous force, etc.

              

  • These damping forces are proportional to the magnitude of the velocity and their direction always opposes the motion.
  • Due to decrease in amplitude the energy of the oscillator also goes on decreasing exponentially
  •  The equation of motion of  Damped oscillation is given by

    m\frac{du}{dt}= -kx-bu

     where 

u=velocity 

  -bu = damping force

b= damping constant

-kx = restoring force

Or using u=\frac{dx}{dt}

where x=displacement of damped oscillation

we can write, The equation of motion of  Damped oscillation as

m\frac{d^2x}{dt}= -kx-b\frac{dx}{dt}

The solution of the above differential equation will give us the formula of x as

 x=A_0e^{-\frac{bt}{2m}}.\sin \left ( \omega' t+\delta \right ) 

where \omega '=angular \ frequency \ of \ the \ damped \ oscillation

and   \omega'= \sqrt{\frac{k}{m}-\left ( \frac{b}{2m} \right )^{2}}= \sqrt{\omega {_{0}}^{2}-\left ( \frac{b}{2m} \right )^{2}}

  • The amplitude in damped oscillation decreases continuously with time according to 

                A=A_{0}.e^{-\frac{bt}{2m}}

  • The energy  in damped oscillation decreases continuously with time according to 

           E=E_{0}.e^{-\frac{bt}{m}}  where E_0=\frac{1}{2}kA_0^2

 

  • Critical damping-  The condition in which the damping of an oscillator causes it to return as quickly as possible to its
    equilibrium position without oscillating back and forth about this position.  

                 Critical damping happens at \omega _0=\frac{b}{2m}

 

Study other Related Concepts

Free, Forced And Damped Oscillation Current Topic

Periodic And Oscillatory Motions
Simple Harmonic Motion (S.H.M.) And Its Equation
Important Terms In Simple Harmonic Motion
Simple Harmonic Motion And Uniform Circular Motion
Composition Of Two SHM
Energy In Simple Harmonic Motion
Oscillations Of A Spring-mass System
Oscillation Of Two Particle System
Oscillation Of Pendulum
Angular Simple Harmonic Motion

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