Simple Harmonic Motion — Question 3

Question 3

A crossing interval is not always half a period

An SHM particle of period $T$ passes $x=A/2$ moving toward positive $x$ at time zero. It next returns to $x=A/2$ moving toward negative $x$.

1. Find the interval between these two crossings.

2. Find the interval from that second crossing to the next positive-direction crossing. Explain why the intervals differ.

Question 3 - Solution

A crossing interval is not always half a period

(a) Derivation and calculation

At $x=A/2$, $\cos\psi=1/2$. The direction condition selects$$v=-A\omega\sin\psi>0\quad\Rightarrow\quad\psi_0=-\pi/3.$$The next equal-position phase is $\psi_1=\pi/3$, hence$$\Delta\psi_1=\frac{2\pi}{3},\qquad\Delta t_1=\frac{\Delta\psi_1}{\omega}=\boxed{\frac T3}.$$

(b) Result and interpretation

The following positive-direction crossing has $\psi_2=5\pi/3$:$$\Delta\psi_2=\frac{5\pi}{3}-\frac\pi3=\frac{4\pi}{3},\qquad\Delta t_2=\boxed{\frac{2T}{3}}.$$$$\Delta t_1+\Delta t_2=T.$$The shorter arc passes through $+A$; the longer arc includes the negative turning point.

Physical check

Only equilibrium crossings divide a sinusoidal cycle into two equal intervals. The stated crossing is displaced from equilibrium.