AIEEE Concepts®: Simple Harmonic Motion

S.H.M.
Simple Harmonic Motion :

The periodic motion, in which a particle moves to and fro about a fixed point such that its acceleration is always directly proportional to its displacement from its mean position, is defined as simple harmonic motion.
Mathematical S.H.M is x = A sin wt
A

amplitude
w =

angular frequency
G = time period.
The motion of a particle which is controlled by the force equation F = - kx is known as simple harmonic motion e.g. motion of simple pendulum ,motion of a liquid column in a U-tube ,motion of a electrical tuning fork etc.

Restoring force :
The force acting on the particle which tends to bring the particle towards its mean position is known as restoring force. This force is always directed towards the mean position
This force always acts in a direction opposite to that of displacement it is given F = - kx and its dimensions are MLT^-2

Phase : The angle (wt + q) is called is the phase of vibration. Phase of a body executing S.H.M at any instant represents its state as regards its position and direction at that instant.

The displacement of the any particle at any instant executing S.H.M given by x = a sin (

t +

)

The velocity v of the oscillation can be obtained by differentiating of (i), we get,
V =

= w

At mean position i.e., at x = 0 the velocity is maximum (wa)
So V_max =

a. The velocity is zero at the extreme positions.

Acceleration, a =

At mean position i.e., at
x =

a = w²A = a_max

A Simple harmonic oscillator possesses kinetic energy as well as potential energy.
Kinetic energy K is given as
K =

Potential energy U is given as
U =

Total energy E = K + U
=

.

If a body of mass 'M' is attached at the free end to a spring of spring constant 'K' and negligible mass and if it is made to oscillate, its time period of oscillation is,

If a spring is hanging vertically and to its free end if a body is attached, if the elongation produced in the spring is 'l' then

If a body of mass 'M' is attached to a spring of mass 'm' and spring constant 'K' and if it is made to oscillate, its time period is

If two springs of spring constants K₁and K₂are connected in series, the effective spring constant K is given by,

If two springs of spring constants K1and K2are connected in parallel as shown in the fig., the effective spring constant (K) is

If a body connected to a spring rolls then the time period if its S.H.M is given by

Where 'K1' is the radius of gyration of the body about an axis passing through its C.G.

A point mass suspended by a massless inextensible string from a rigid support constitute a simple pendulum.
The time period of simple pendulum is given by :

Note : The pendulum which passes a time of two second is known as second's pendulum.

If bob of a simple pendulum is made to oscillate in a fluid of density