Explain
(i) Undamped and damped oscillations and
(ii) free, forced and resonant oscillations with illustrations.

Explanation

A body set into oscillation, continues to oscillate for ever with the same amplitude if damping forces are not acting, because mechanical energy is conserved. If damping forces are present, the amplitude of oscillation gradually decreases because energy is dissipated. Such oscillations are called damped oscillations. In most cases, the damping force is directly proportional to the speed e.g. viscous drag due to air. Thus damping force is -bv,with displacement x and velocity `v=(dx)/(dt)` is `F_(net)=-kx-bv`
Newton's second law gives -kx-bv=ma or `"m"(d^(2)x)/(dt^(2))+"b"(dx)/(dt)+kx=0`
where, b is damping constant
The solution for this equation is `x(t)=A_(0)e(b)/(2m)tcos (omegat+delta)`
`omega= sqrt(omega_(0)^(2)-((b)/(2m))^(2)) ` delta is phase constant.
where,`omega_(0)=sqrt((k)/(m))`, the angular frequency for the case with no damping)