Simple Harmonic Motion and Uniform Circular Motion — Question 3

Question 3

Changing the viewing axis changes phase rather than amplitude

A point has coordinates $x=A\cos\omega t$ and $y=A\sin\omega t$. An observer measures its signed projection $u$ on a fixed axis making angle $\alpha$ counterclockwise from the positive $x$ axis.

1. Derive $u(t)$ and state its amplitude, period and phase relative to $x(t)$.

2. For $\alpha=\pi/3$, find $u(0)$ and $\dot u(0)$. Does measuring on an oblique axis reduce the full projection amplitude?

Question 3 - Solution

Changing the viewing axis changes phase rather than amplitude

(a) Derivation and calculation

$$\hat{\mathbf e}_u=(\cos\alpha,\sin\alpha),\quad u=\mathbf r\cdot\hat{\mathbf e}_u.$$$$u=A\cos\omega t\cos\alpha+A\sin\omega t\sin\alpha=\boxed{A\cos(\omega t-\alpha)}.$$$$\boxed{A_u=A},\qquad\boxed{T=2\pi/\omega},\qquad\phi_u=-\alpha.$$

(b) Result and interpretation

For $\alpha=\pi/3$,$$\boxed{u(0)=A/2},\qquad\boxed{\dot u(0)=A\omega\sin\alpha=\sqrt3A\omega/2}.$$$$\max_t|\mathbf r\cdot\hat{\mathbf e}_u|=A.$$During a complete rotation the vector aligns with every fixed viewing axis, so the full amplitude remains $A$.

Physical check

The initial projection can be smaller than $A$ without reducing the maximum over a cycle.