What is damped oscillation. Write the differential equation for damped oscillations. Obtain an expression for the displacement in the case of damped oscillatory motion. Discuss how the amplitude of oscillation changes with time?

Consider a loaded spring oscillating in the vertical direction in a resistive media.
Let m be the mass of load and k be the spring constant.
The load is displaced from the equilibrium position. Let at any instant, y be the displacement from equilibrium position and v is the velocity of the load.
The different forces acting on load are:

(i) Restoring force which is proportional to displacement and directed towards equilibrium position.
i.e.,    F₁= – ky

(ii) Resistive force of medium which is proportional to velocity and opposite to the direction of velocity.
i.e.,     F₂ = –bv
The net force acting on the load is,

F = F_x1 + F₂ 
   = – ky –bv
Therefore the equation of motion of load is, 

This is the required differential equation of motion of damped oscillations.
The solution of the equation is,
 
Comparing it with , we get,
                                      .....(1)

Equation (1) gives the amplitude of oscillation, which decreases exponentially with time.

Equation (2) gives the frequency of oscillation which is less than the natural frequency of oscillation ω₀.