# How do I solve for the phase constant given the amplitude and the angular frequency?

A piston (with mass M) in a car engine is in vertical simple harmonic motion with amplitude A. The engine is running at a period T. Suppose a small piece of metal with mass m were to break loose from the surface of the piston when it (the piston) is at the lowest point z and velocity is v. At what position in meters does the piece of metal lose contact with the piston?

Also the maximum velocity and acceleration are given.

The text book gives these equations:

$x(t) = Acos(wt+φ)$

$v(t) = -wAsin(wt+φ)$

$a(t) = -w^2Acos(wt+φ)$

So I propose that the best way to figure this out would be to solve for the time at which acceleration is at its max, then plug that time into $x(t)$. But I must first solve for the phase constant. Is this correct? If so, how do I find the phase constant?

The phase constant is needed only if you have a specific initial condition, e.g. if I told you where $x$ was at time $t = 0$, you could solve for $\varphi$. Otherwise you can just choose whatever you want for it: Note that it is the same in all functions.
Choosing some value for $\varphi$ is analogous to you manually setting the time origin to something you like. Imagine observing the harmonic motion for a while, and then deciding that $t = 0$ should be the time where the piston is all the way up. Or you could decide that $t = 0$ should be the time where the piston is all the way down. Or right in the middle.
Thus, for your problem, you can just set $\varphi = 0$.