Give the expression for the amplitude of a damped oscillation of a particle. Hence discuss the amplitude for driving frequency (a) far from natural frequency and (b) close to natural frequecny.

Give the expression for the amplitude of a damped oscillation of a par

1.	Give the expression for the amplitude of a damped oscillation of a particle. Hence discuss the amplitude for driving frequency (a) far from natural frequency and (b) close to natural frequecny.
Answer» Solution :Let a time dependent `F(t)=F_(0)cos omega_(d)t`. For the motion of a particle under the combined action of forces such as linear RESTORING FORC, damping force and time dependent driving force. `"i.e."F=F_(t)+F_(d)+F(t)` `"i.e."ma(t)=-kx(t)-bv(t)+F_(0) cos omega_(d)t.` where `x(t)=A cos(omegadt+phi)`. The amplitude A is a function of the forced frequency `omega_(d)` and natual frequency `omega.` `"i.e"A=(F_(0))/({m^(2)(omega^(2)-omega_(0)^(2))^(2)+omega_(d)^(2)b^(2)}^(1//2))` (a) For driving frequency far from natural frequency, the `omega_(d)b lt lt m (omega^(2)-omega_(d)^(2))` `therefore A=(F_(o))/(m(omega^(2)-omega_(d)^(2)))` for `omega=omega_(d), A rarroo` for zero damping (ideal case) (b) For diving frequency, close to natural frequency. `m(omega^(2)-omega_(d)^(2))lt lt omega_(d)b` `A=(F_(0))/(omega_(d)b)` for `omega_(d)` very close to `omega.` Hence maximum possible amplitude DEPENDS on the driving frequecny `A prop (1)/(omega_(d))`. Hence resonance is the condition of OSCILLATOR whose amplitude becomes maximum as the driving frequency approaches the natural frequency.