A particle of mass `m` can perform undamped harmonic oscillations due to an electric force with coefficient `k`. When the particle was in equilibrium, a permanent force `F` was applid to to its for `tau` seconds. Find the oscillation amplitude that the particle acquired after the action of the force ceased. Draw the approximate plot `x(t)` of oscillations. Investigate possible cases.

Explanation

We have to look for solutions of the equation
`m ddot(x) + kx=F,0 lt t_(1) lt x`
`mddot(x)+kx=0, t gt tau`
subject to `x(0)=dot(x)(0)=0` where `F` is constant.
The solution of this equation will be sought in the form
`x=(F)/(k)+A cos ( omega_(0)t+ alpha), 0 le t le tau`
`x=B cos ( omega_(0)(t-tau)+ beta), t gt tau`
`A` and `tau` will be determined from the baoundary condition at `t=0`.
`0=(F)/(k)+A cos alpha`
`0=- omega_(0)Asin alpha`
Thus, `alpha=0` and `A=-(F)/(k)` and `x=(F)/(k)(1- cos omega_(0)t) 0 le t lt tau.`
`B` and `beta` will be determiing by the continuity of `x` and `dot(x)` at `t= tau`. Thus `(F)/(k) ( 1- cos omega_(0)tau)=B cos beta` and `cancel(omega_(0))(F)/(k) sin omega_(0) tau=- cancel( omega_(0))B sin beta`
Thus `B^(2)=((F)/(k))^(2)(2-2cos omega_(0)tau)`
or `B=2 (F)/(k) | sin ( (omega_(0)tau)/(2))|`