Circular Motion and Polar Coordinates

Uniform Circular Motion

For motion in a circle of radius $r$ at constant speed $v$:

Centripetal acceleration: $a_c = \frac{v^2}{r} = \omega^2 r$

Angular velocity: $\omega = \frac{v}{r}$

Period: $T = \frac{2\pi r}{v} = \frac{2\pi}{\omega}$

Polar Coordinates

Position in polar coordinates: $(r, \theta)$

Unit vectors: $\hat{r}$ (radial), $\hat{\theta}$ (tangential)

Important: These unit vectors are not constant—they change direction as the particle moves.

Time derivatives of unit vectors: $\frac{d\hat{r}}{dt} = \dot{\theta}\hat{\theta} = \omega\hat{\theta}$

$\frac{d\hat{\theta}}{dt} = -\dot{\theta}\hat{r} = -\omega\hat{r}$

Velocity in Polar Coordinates

Position vector: $\vec{r} = r\hat{r}$

Velocity: $\vec{v} = \frac{d\vec{r}}{dt} = \frac{dr}{dt}\hat{r} + r\frac{d\hat{r}}{dt}$

$\vec{v} = \dot{r}\hat{r} + r\dot{\theta}\hat{\theta}$

Radial component: $v_r = \dot{r}$

Tangential component: $v_{\theta} = r\dot{\theta} = r\omega$

Speed: $v = \sqrt{v_r^2 + v_{\theta}^2} = \sqrt{\dot{r}^2 + r^2\dot{\theta}^2}$

Acceleration in Polar Coordinates

$\vec{a} = \frac{d\vec{v}}{dt} = \frac{d}{dt}(\dot{r}\hat{r} + r\dot{\theta}\hat{\theta})$

After applying product rule and substituting unit vector derivatives:

$\vec{a} = (\ddot{r} - r\dot{\theta}^2)\hat{r} + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{\theta}$

Radial component: $a_r = \ddot{r} - r\omega^2$

Tangential component: $a_{\theta} = r\alpha + 2\dot{r}\omega$

where $\alpha = \ddot{\theta}$ is angular acceleration.

Circular Motion (r = constant)

When radius is constant ($\dot{r} = 0$, $\ddot{r} = 0$):

Radial acceleration (centripetal): $a_r = -r\omega^2 = -\frac{v^2}{r}$

Tangential acceleration: $a_{\theta} = r\alpha = r\frac{d\omega}{dt}$

The negative sign in $a_r$ indicates the acceleration points toward the center (opposite to $\hat{r}$).

Non-Uniform Circular Motion

When speed changes: both centripetal and tangential acceleration exist.

Centripetal: changes direction of velocity Tangential: changes magnitude of velocity

Total acceleration magnitude: $a = \sqrt{a_c^2 + a_t^2} = \sqrt{\left(\frac{v^2}{r}\right)^2 + (r\alpha)^2}$

Example: Spiral Motion

A particle moves in a spiral: $r(t) = bt$, $\theta(t) = \omega t$ (both constant $b$ and $\omega$)

Position: $\vec{r} = bt\hat{r}$

Velocity: $\vec{v} = b\hat{r} + bt\omega\hat{\theta}$

Acceleration: $\vec{a} = -bt\omega^2\hat{r} + 2b\omega\hat{\theta}$

Notice the $2\dot{r}\omega$ term (Coriolis-like term) appears even though $\alpha = 0$.