Seminar Report on
Simple Harmonic Motion
The Linear Differential Equation with constant coefficients find their
most important applications in the study of electrical, mechanical and
other linear systems, especially oscillatory systems.
What is Simple Harmonic Motion?
When the acceleration of a particle is proportional to tis displacement
from a fixed point and is always directed towards it, then the motion
is said to be Simple Harmonic.
Case 1: Spring-Mass Systems
Consider a mass mis attached to the free endof a mass-less spring.
x=0 is the mean position; B depicts compression and C, elongation by
a distance ‘x’. Let the maximum displacement be ‘A’-Amplitude
The spring force is given by:
F(x)= -kx where ‘k’ is the Spring constant.
By force balance, -kx=ma
x + kx = 0
x=0 where µ2
Solution of the differential equation is x=c1cos µt + c2sin µt
Velocity ‘v’= dx = µ(-c1sinµt+c2cosµt)
at t=0, x=a => c1=A
at t=0, v=0 =>c2=0
Therefore x=A cos µt
From the above equation of motion, we can find the following:
Time period of oscillation T=2π = 2π√(m)
frequency = T-1
Maximum velocity = Aµ
Maximum acceleration= Aµ2