Question

A particle of mass (m) is attached to a spring (of spring constant k) and has a narural angular frequency omega_(0). An external force `R(t)` proportional to cos omegat(omega!=omega)(0) is applied to the oscillator. The time displacement of the oscillator will be proprtional to.

Answer 1

Given the natureal angular frequency `=omega_(0)`.
If the displacement of the particle is y, then acceleration of the particle is, `a_(0)=-omega_(0)^(2)y`
The external force `F(t)propcosomegat.` It has an angular frequency `omega`. For the displacement y, the acceleration produced by this force is
`a_(1)=omega^(2)y`
Net acceleration of the particle at displacement y is
`a=a_(0)+a_(1)=-omega_(0)^(2)y+omega^(2)y=-(omega_(0)^(2)-omega^(2))y`
Net force on the particle at displacement y is
`F=ma=-m(omega_(0)^(2)-omega^(2))y`
or `y=(F)/(m(omega_(0)^(2)-omega^(2)))`
So`yprop (1)/(m(omega_(0)^(2)-omega^(2)))`