At the moment `t=0` a particle of mass m starts moving due to a force `F=F_0 cos omegat`, where `F_0` and `omega` are constants. How long will it be moving until it stops for the first time? What distance will it traverse during that time? What is the maximum velocity of the particle over this distance?

Explanation

According to the problem, the force acting on the particle of mass m is, `vecF=vecF_0 cos omegat`
So, `m(dvecv)/(dt)=vecF_0cos omegat` or `dvecv=(vecF_0)/(m)cos omega tdt`
Integrating, within the limits.
`underset(0)overset(vecv_0)int dvecv=(vecF_0)/(m)underset(0)overset(t)intcos omegadt` or `vecv=(vecF_0)/(momega)sin omegat`
It is clear from equation (1), that after starting at `t=0`, the particle comes to rest from the first time at `t=pi/omega`.
From Eqs. (1), `v=|vecv|=(F_0)/(momega)sin omegat` for `tlepi/omega` (2)
Thus during the time interval `t=pi//omega`, the sought distance
`s=(F_0)/(mw)underset(0)overset(pi//omega)int sin omega t dt=(2F)/(momega^2)`
From Eq. (1)
`v_(max)=(F_0)/(momega)` as `|sin omega t|le1`