Let the equation of motion of two particles be, 

where,

ϕ is the phase difference between two particles.

Each time the particles cross at a distance 0.707A.

Therefore, 

          0.707A=A sin 

and    0.707A=A sin(wt+) 

       0.707=sin and, 

          0.707= sn 

Now,

     sin(=sin(=sin 

     

Since,  

 

We can see that ϕ = 0 is not possible.

Therefore the phase difference between two particles is 90°. 

When an oscillator is displaced from equilibrium position, it executes simple harmonic motion. The time period of its oscillation depends upon its inertia factor and elastic factor. If the amplitude of oscillations remains constant in the absence of any external source of energy, then the oscillations are called free oscillations and oscillator is called free oscillator. The frequency with which the oscillator oscillates is called natural frequency.

The differential equation of free oscillator is,



The solution of the above equation is,



The energy of oscilation is,



As the amplitude of free oscillations is constant, therefore energy of oscillation is also constant. The energy-time and displacement-time curve for free oscillation is as shown in figure.

When an oscillator oscillates in resistive medium, the energy of oscillation is consequently decreased. The amplitude of such an oscillator decreases with time and ultimately the oscillations die out. This type of oscillator is called damped oscillator.

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

(i) the 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 differntial 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.