Question

A ball of mass $m$ can perform damped harmonic oscillations about the point $x=0$ with natural frequency ${\omega }_0$. At the moment $t=0$, when the ball was in equilibrium, a force $F_x=F_0\cos \omega t$ coinciding with the $x$ axis was applied to it. Find the law of forced oscillation $x\left(t\right)$ fro that ball.

Answer 1

The equation of motion of the ball is
$m(\ddot{x}+{\omega }_0^{2}x)=F_0\cos \omega t$
This equation has the solution
$x=A\cos ({\omega }_{0}t+\alpha )+B\cos \omega t$
where $A$ and $\alpha$ arte arbitrary and $B$ is obtained by substitution in the above equation
$B=\frac{F_0/m}{{\omega }_0^2-{\omega }^2}$
The conditions $x=0,\dot{x}=0$ at $t=0$ give
$A\cos \alpha +\frac{F_0/m}{{\omega }_0^2-{\omega }^2}=0$ and $-{\omega }_{0}A\sin \alpha =0$
This gives $\alpha =0,A=-\frac{F_0/m}{{\omega }_0^2-{\omega }^2}=\frac{F_0/m}{{\omega }_0^2-{\omega }^2}$
Finally $x=\frac{F_0/m}{{\omega }_0^2-{\omega }^2}(\cos {\omega }_{0}t-\cos \omega t)$