15-87.

The point of this problem is to give some practice in evaluating the two constants $A$ and $\varphi$ given an initial condition. We have the equations

$$\begin{array}{ccc\vert ccc} x(t) &=& A\cos(\omega t +\varphi... ...& A\cos(\varphi) & v(0) &=& -\omega A\sin(\varphi), \end{array}$$

from which we can evaluate the initial conditions. Now, one condition is that $V(0)=0$, which yields

$$0=\omega A\sin(\varphi),$$

so either $\omega$, $A$, or $\sin(\varphi)$ is zero. Our oscillator would be boring if either the first two were true, so we go with the last one, which means either $\varphi=0$ OR $\varphi=\pi$. Make sure you realize the angle can have multiple values!

Next, we use the position condition

$$x(0)=.37=A\cos(\varphi).$$

Since we (usually) take $A$ to be a positive quantity, we need $\varphi=0$, since otherwise $\cos$ comes out negative. But, $\cos(0)=1$, so $A=.37$.

Note, we could have used $\varphi=\pi$ and $A=-.37$. Either is fine.