A ball of mass `m` can perform undamped harmonic oscillations about the point `x=0` with natural frquency, `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(t)` for that ball.

Explanation

The equation of motion of the ball is
`m(ddot(x) + omega_(0)^(2)x)=F_(0)cos omegat `
This equation has the solution
`x=A cos ( omega_(0)t + alpha)=Bcos omegat `
Where `A` and `alpha` are arbitrary and `B` is obtained by substitution in the above equation
`B=( F_(0)//m)/(omega_(0)^(2)-omega^(2))`
The conditions `x=0, dot(x) =0` at `t=0 ` give
`A cos alpha +(F_(0)//m)/(omega_(0)^(2)-omega^(2))=0` and `- omega_(0) A sin alpha=0`
This gives `alpha=0`, `A=-(F_(0)//m)/(omega_(0^(2))-omega^(2))=(F_(0)//m)/(omega^(2)-omega_(0)^(2))`
Finally `x=(F_(0)//m)/(omega^(2)-omega_(0)^(2))( cos omega_(0)t-cos omegat)`