What is Simple Harmonic Motion (SHM)

Definition:

                A special type of vibratory motion in which of the body is directly proportional to the displacement from the mean position and of the body is always directly towards the mean position are called Simple Harmonic Motion.

Mathematically:

a ∝ -x

Mass spring system:

 Let we have a mass attached to a spring

    If the spring is stretched and compressed through a small displacement from its mean position.

According to Hooke’s law:

F_ext ∝ k x

The external force is directly proportional to the displacement within the elastic limit.

F_ext = k x
k = spring constant

    Unit of k is Nm^-1

F_res ∝ - x

F_res = - k x

m a = - k x

a = - x

a ∝ -x

Hence mass spring system performs Simple Harmonic Motion.

Characteristics:

• Ø Velocity:
  

As the body goes away from the mean position its velocity goes on decreasing and as the body goes towards the mean position, its velocity goes on increasing.

Velocity is maximum at mean position and velocity is minimum at extreme position.

From O to A, body is going away from the mean position. So, velocity is decreases

From A to O, body is going towards mean position. So, velocity is increases.

• Ø Acceleration:

As the body goes away from the mean position its acceleration goes on increasing and as the body goes towards the mean position, acceleration goes on decreasing.

Acceleration is maximum at mean position and acceleration is minimum at extreme position.

From O to A, body is going away from the mean position. So, acceleration is increase.

From A to O, body is going towards mean position. So, acceleration is decrease.

• Ø Energy:

 As the body goes away from the mean position its velocity goes on decreasing so kinetic energy also decrease and as the body goes towards the mean position, its velocity goes on decreasing so kinetic energy also increase

As the body goes away from the mean position its displacement goes on decreasing so potential energy also decrease and as the body goes towards the mean position, its displacement goes on increasing so potential energy also increase

Velocity and kinetic energy are maximum at mean position.

Displacement and potential energy are minimum at extreme position.

Velocity and kinetic energy are minimum at mean position.

Displacement and potential energy are maximum at extreme position.

From O to A, body is going away from the mean position. So, velocity and kinetic energy are decreases or displacement and potential energy are increases.

From A to O, body is going away from the mean position. So, velocity and kinetic energy are increases. or displacement and potential energy are decreases.

• Ø Time period:

Definition:

Ø Time to complete one revolution about mean position is called Time period.

Unit:

Sl unit is second (s)

• Ø Frequency:

Definition:

The number of  one revolution per second is called frequency.

Unit:

Sl unit is per second (s^-1) or Hertz (Hz)