Simple Harmonic Motion Question

An object in simple harmonic motion passes through the central point $x=0$ at time $t=0$ and every second thereafter.

Find an equation for the motion if$$v(0)=-4.$$

Because the particle passes through the central point $x=0$ when $t=0$, it is natural to use a sine function:$$x=A\sin(\omega t).$$

The particle passes through the central point every second.

However, passing through the centre occurs twice during each complete cycle — once moving in one direction and once moving in the other.

Therefore, the time from one central crossing to the next is half a period:$$\frac{T}{2}=1.$$

Hence,$$T=2\text{ seconds}.$$

For simple harmonic motion,$$\omega=\frac{2\pi}{T}.$$

Since $T=2$,$$\omega=\frac{2\pi}{2}=\pi.$$

Our equation is therefore$$x=A\sin(\pi t).$$

We still need to determine $A$.

For simple harmonic motion,

When and

Hence,