Question

Deduce differential equation of S.H.M &hence obtain the expression for acceleration velocity & displacement .Also state their max & min values​

Answer 1

Answer:

In any SHM, there always exists a restoring force which tries to bring the object back to mean position.

This force causes acceleration in the object.

Hence F=−kx.   Here F is the restoring force, x is the displacement of the object from the mean position, and k is the force per unit displacement.

The −Ve sign indicates that the force is opposite to the displacement. Only then can the object be brought back after displacement.

We know that by Newton's 2nd law of motion, F=ma    ∴a=mF

Applying this, we get a=−mkx

Let mk be ω2 ------ω is angular frequency.

Hence we have a=−ω2x                     expression for acceleration

In calculus a=dt2d2x ----------the double derivative of displacement with respect to time.

dt2d2x+ω2x=0                   the differential equation

We can also write a=dtdv---------derivative of velocity w.r.t. time is acceleration.

∴dtdv=−ω2x

dtdv=dxdv×dtdx=vdxdv                      ∵dtdx=v

∴vdxdv=