Show that the equation `x=Acos(omegat-alpha)` is a mathematical representation of simple harmonic motion. Find the time period and the maximum speed of the motion.

Explanation

`x=Acos(omegat-alpha)`.
`therefore` Velocity, `v=dx/dt=-Aomegasin(omegat-alpha)`
The magnitude of `sin(omegat-alpha)` is maximum when it is `pm1`. This corresponds to the maximum velocity, or simply, the maximum speed.
`therefore` Maximum speed = `pmAomega`.
Again, `cos(omegat-alpha)=cos(omegat-alpha+2pi)`. So, a particle executing SHM returns to its initial phase when the phase angle changes by `2pi`. The time elapsed between them is the time period (T) of the SHM.
`therefore" "omega(t+T)-alpha=omegat-alpha+2pi`
or, `omegaT=2pior,T=(2pi)/omega`.
Now, acceleration, `a=(dv)/(dt)=-Aomega^(2)cos(omegat-alpha)`,
i.e., `a=-omega^(2)x`
This relation signifies that the given equation is a mathematical representation of SHM.