Free, Forced And Damped Oscillation - Practice Questions & MCQ

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{d u}{d t}=-k x-b u
  $

  where
  $\mathrm{u}=\mathrm{velocity}$
  $-b u=$ damping force
  $\mathrm{b}=$ damping constant
  $-k x=$ restoring force

  Or using

  $
  u=\frac{d x}{d t}
  $

  where $\mathrm{x}=$ displacement of damped oscillation
  we can write, The equation of motion of Damped oscillation as

  $
  m \frac{d^2 x}{d t}=-k x-b \frac{d x}{d t}
  $

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

  $
  x=A_0 e^{-\frac{b t}{2 m}} \cdot \sin \left(\omega^{\prime} t+\delta\right)
  $

  where $\omega^{\prime}=$ angular frequency of the damped oscillation

  $
  \text { and } \omega^{\prime}=\sqrt{\frac{k}{m}-\left(\frac{b}{2 m}\right)^2}=\sqrt{\omega_0^2-\left(\frac{b}{2 m}\right)^2}
  $

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

  $
  A=A_0 \cdot e^{-\frac{b t}{2 m}}
  $
   

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

  $
  E=E_0 \cdot e^{-\frac{b t}{m}} \text { where } E_0=\frac{1}{2} k A_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}{2 m}
  $