Chapter 12, Problem 31RE

As illustrated in the accompanying figure on the next page, suppose that a particle moves counterclockwise around a circle of radius R centered at the origin at a constant rate of $\omega$ radians per second. This is called uniform circular motion. If we assume that the particle is at the point $\left(R,0\right)$ at time $t=0,$ then its position function will be $\text{r}\left(t\right)=R\cos \omega t\text{i}+R\sin \omega t\text{j}$

(a) Show that the velocity vector $\text{v}\left(t\right)$ is always tangent to the circle and that the particle has constant speed $\upsilon$ given by $\upsilon =R\omega$

(b) Show that the acceleration vector $\text{a}\left(t\right)$ is always directed toward the center of the circle and has constant magnitude a given by $a=R{\omega }^2$

(c) Show that the time T requires for the particle to make one complete revolution is $T=\frac{2\pi }{\omega }=\frac{2\pi R}{\upsilon }$