1. Differential Equation of the Mechanical Oscillator
Prepared by;
Dr. Rajesh Sharma
Assistant Professor
Dept of Physics
P.G.G.C-11, Chandigarh

2. Substituting this in eq.(1), we get
The Hook’s law
(1)
Let there be a body of mass m attached to a spring. Then according to the Newton’s Second law of motion, we have
Substituting this in eq.(1), we get
Or, (2)
Putting where w0 is another constant for the oscillator, we get
Or, (3)
This is called the Differential Eq. of SHM and its solution is of the form
Where A is the amplitude and w0 = S/m; T=2 m/s

3. Energy of the Mechanical Oscillator
The particle executing SHM possesses Kinetic as well as Potential energies.
KINETIC ENERGY:
The KE of the body of mass m, when possessing the velocity v=dx/dt is given by
(4)
Since and 
Or, (5)

4. POTENTIAL ENERGY:
The particle executing SHM is moved against the restoring force and the work so done is stored as the potential energy. Let us displace the particle by dx , then the work done which is equal to the potential energy stored in the system is given by:
Here the –ve sign indicates the work done against the restoring force. Now, for the mass attached to the spring, we have
We have
The potential energy of the oscillator, when the displacement is x, becomes
Or, (6)
Putting we get
(7)

5. Total energy of the oscillator
The total energy of the oscillator is given by
Or,
But, therefore, Hence,
Or, (8)
This shows that the total energy of the mechanical oscillator is constant or is conserved and is independent of the location of the particle. I depends upon m, A, w0 or S (because w0 = S/m).

6. Average Kinetic Energy
From Eq.(5), we have
The Average KE over the time period is given by
Therefore

7. Average Potential Energy
The potential energy
Putting we get
The Average potential energy over the time period is given by

8. Also, the total energy of the oscillator is
Which shows that
That is, the average potential energy is equal to the average kinetic energy over a period and is equal to half the total energy of the mechanical oscillator.