Question

A particle of mass $m$ is attached to a spring (of spring constant $k$) and has a natural angular frequency ${\omega }_0$. An external force $F\left(t\right)$ proportional to $\cos \omega t,\omega \ne {\omega }_0$ is applied to the oscillator. The time displacement of the oscillator will be proportional to:

• A. $\frac{m}{{\omega }_0^2-{\omega }^2}$
• B. $\frac{1}{m\left({\omega }_{0^{-}}^2{\omega }^2\right)}$
• C. $\frac{1}{m({\omega }_o^2+{\omega }^2)}$
• D. $\frac{m}{{\omega }_o^2}$

Answer 1

The correct option is B $\frac{1}{m\left({\omega }_{0^{-}}^2{\omega }^2\right)}$

For forced oscillations, displacement is given by:
$x=Asin(\omega t+\varphi )$
where $A=\frac{F_0/m}{{\omega }_0^2-{\omega }^2}$
Therefore, $x\propto \frac{1}{m({\omega }_0^2-{\omega }^2)}$