Chapter 12, Problem 6MC

Suppose that the position function of a point moving in the xy plane is $\text{r}=x(t)\text{i}+y(t)\text{j}$

This equation can be expressed in polar coordinates by making the substitution

$x(t)=r(t)\cos \theta (t),y(t)=r(t)\sin \theta (t)$

This yields

$\text{r}=r(t)\cos \theta (t)\text{i}+r(t)\sin \theta (t)\text{j}$

which can be expressed as

$\text{r}=r(t)e_r(t)$

where $e_r(t)=\cos \theta (t)\text{i}+\sin \theta (t)\text{j}$.

a. Show that ${\text{e}}_r(t)$ is a unit vector in the same direction as the radius vector $\text{r}$ if $r(t)>0$. Also, show that ${\text{e}}_{\theta }(t)=-\sin \theta (t)\text{i}+\cos \theta (t)\text{j}$ is the unit vector that results when ${\text{e}}_r(t)$ is rotated counterclockwise through an angle of $\pi /2$. The vector ${\text{e}}_r(t)$ is called the radial unit vector and the vector ${\text{e}}_{\theta }(t)$ is called the transverse unit vector (see the accompanying figure).

b. Show that the velocity function $\text{v}=\text{v}(t)$ can be expressed in terms of radial and transverse components as

$v=\frac{dr}{dt}{e}_r+r\frac{d\theta }{dt}{e}_{\theta }$

c. Show that the acceleration function $\text{a}=\text{a}(t)$ can be expressed in terms of radial and transverse components as

$a=\left[\frac{d^{2}r}{d{t}^2}-r{\left(\frac{d\theta }{dt}\right)}^2\right]e_r+\left[r\frac{d^2\theta }{d{t}^2}+2\frac{dr}{dt}\frac{d\theta }{dt}\right]e_{\theta }$

FIGURE Ex-6