Derive the differential equation for the Linear S.H.M
Answers
when a particle performs linear SHM the force acting on the particle is directed towards the mean position and its magnitude is directly proportional to the displacement of a particle from the mean position
F = - k x
(restoring force)
here ,
- k=force constant
- x=distance of the particle from the mean position
By Newton's second law of motion,
F = ma
v= d x / d t
a = d v / d t
substituting v in a
a= d (d x) / d t (d t)
a = d ² x / d t ²
F = ma = - k x
m d ² x / d t ² = - k x
d ² x / d t ² = - k x / m
d ² x / d t ² + k x / m = 0
we know that ,
w = √ k/ m
w² = k / m
substituting k / m as w ²
d ² x / d t ² + w² x = 0
Answer:
The differential equation for the Simple harmonic motion has the following solutions: x = A sin ω t x=A\sin \omega \,t x=Asinωt (This solution when the particle is in its mean position point (O) in figure (a)