Circular Motion

An object that moves in a circle at a constant speed is said to be experiencing a uniform circular motion. The magnitude of the velocity is constant but the direction is changing. As the direction changes, so too must velocity, as velocity is a measure of rate of displacement along a direction. Therefore, an object experiencing a uniform circular motion must also be experiencing a constant acceleration.

$$\vec{a}_{average} = \frac{\vec{v_2}-\vec{v_{1}}}{\Delta t} = \frac{\Delta\vec{v}}{\Delta t}$$ If we were to take an infinitesimal $\Delta t$, $\Delta \vec{v}$ approaches the center of the circle in direction.

For uniform circular motion, $v_{1}$ and $v_{2}$ are of the same magnitude, $v$.

$$\frac{\Delta v}{v} \approx \frac{l}{r} \therefore \Delta v \approx \frac{lv}{r}$$

If we return to our definition for acceleration:

$$\vec{a} = \frac{\Delta \vec{v}}{\Delta t} = \frac{v}{r} \frac{l}{\Delta t} $$

$$\frac{l}{\Delta t} = v$$

$$\therefore a_{r} = \frac{v^2}{r}$$

Centripetal / Radial Acceleration

Therefore, in uniform circular motion, an object is constantly accelerating. The direction of this acceleration is towards the center of the circle, along $r$.

$$a_{r} = \frac{v^2}{r}$$

$v$ is the linear velocity of the object. This velocity has a constant magnitude.

$r$ is the radius of the circular motion.

$a_{r}$ and $v$ are perpendicular.

Period and Frequency

The period $T$ is the time required to complete one revolution around the circle. It has units of seconds (s).

The frequency $f$ is the number of revolutions per unit of time, often measured in Hertz (Hz), where 1 Hz = 1 revolution per second.

The relationship between period and frequency is given by:

$$f = \frac{1}{T}$$

The linear velocity of the object is related to its period $T$ by:

$$v = \frac{\text{circumference}}{T} = \frac{2\pi r}{T}$$

Radial Acceleration in Terms of Angular Velocity

The radial acceleration $a_r$ can also be expressed in terms of the angular velocity $\omega$, which is the rate at which the angle changes as the object moves around the circle:

$$a_r = r \omega^2$$

This equation shows that the radial acceleration depends on both the radius of the circular path and the angular velocity.

Angular Velocity and Linear Velocity Relationship

The linear velocity $v$ of an object moving in a circle is related to its angular velocity $\omega$ by the equation:

$$v = r \omega$$

Where:

$v$ is the linear velocity.
$r$ is the radius of the circle.
$\omega$ is the angular velocity.

Angular Displacement

The angular displacement $\Delta \theta$ represents the angle through which an object moves along a circular path.

The average angular velocity $\omega_{avg}$ is defined as:

$$\omega_{avg} = \frac{\Delta \theta}{\Delta t}$$

Where:

$\Delta \theta$ is the change in angle.
$\Delta t$ is the change in time.

For instantaneous angular velocity, we take the limit of $\Delta t$ approaching zero:

$$\omega = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t}$$

Summary of Key Relationships

Radial (centripetal) acceleration:

$$a_r = \frac{v^2}{r}$$

or equivalently,

$$a_r = r \omega^2$$
Linear velocity in terms of angular velocity:

$$v = r \omega$$
Frequency and period:

$$f = \frac{1}{T}$$

$$v = \frac{2\pi r}{T}$$
Angular velocity:

$$\omega = \frac{\Delta \theta}{\Delta t}$$
Relationship between period and frequency:

$$T = \frac{1}{f}$$

$$v = \frac{2\pi r}{T}$$