Simple Harmonic Motion and Uniform Circular Motion — Question 1

Question 1

A projected coordinate retains the direction through its velocity

A point moves counterclockwise around a circle of radius $A=.10\,\mathrm{m}$ with angular speed $\omega=4\,\mathrm{rad/s}$. At $t=0$ its polar angle is $\pi/3$. Its horizontal projection is $x$.

1. Write $x(t)$ and find the initial projected velocity and acceleration.

2. Compare the projected speed with the actual speed at that instant. Is the projected acceleration always equal in magnitude to the centripetal acceleration?

Question 1 - Solution

A projected coordinate retains the direction through its velocity

(a) Derivation and calculation

$$\boxed{x(t)=.100\cos(4t+\pi/3)\,\mathrm m}.$$$$v_x=-A\omega\sin\psi,\quad a_x=-A\omega^2\cos\psi=-\omega^2x.$$$$\boxed{v_x(0)=-.400\sin(\pi/3)=-.34641\,\mathrm{m/s}},$$$$\boxed{a_x(0)=-1.60\cos(\pi/3)=-.800\,\mathrm{m/s^2}}.$$

(b) Result and interpretation

$$v_{\mathrm{circle}}=A\omega=.400\,\mathrm{m/s},\quad\frac{|v_x(0)|}{v_{\mathrm{circle}}}=\frac{\sqrt3}{2}.$$$$a_c=A\omega^2=1.60\,\mathrm{m/s^2},\quad |a_x|=a_c|\cos\psi|\le a_c.$$Acceleration magnitudes agree only at $x=\pm A$; the projected acceleration is zero at the center.

Physical check

The projection changes magnitude and direction even though the circle has constant speed.