Question

Derive the differential equation for the Linear S.H.M ​

Answer 1

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 2

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)