Circular Motion

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For CSEC Physics and this section of CAPE Physics Unit 1 so far, we have been studying linear motion. You will find that most concepts in linear motion have an equivalent in circular motion - the study of motion along curves.

Angular velocity ($\omega$)

This is known as angular frequency. It is the rate of change of angular displacement($\theta$): $$\omega = \frac{\theta}{t}$$

Recall that the arc length($s$) of a circle when the angle is given in radians is: $$s=\theta r$$

If we divide both sides by time: $$ \begin{equation}\begin{aligned} \frac{s}{t}&=\frac{\color{green}\theta \color{normal}r}{\color{green}t\color{normal}}\\ v&=\color{green}\omega \color{normal}r\\ \end{aligned}\end{equation} $$

The linear velocity($v$) of the object is the product of its angular velocity($\omega$) and the radius($r$) of the circular path in which the object is moving.

If we apply the rate of change formula($\omega=\frac{\theta}{t}$) above to an entire revolution, the value of $\theta$ will be $2\pi$ and that of $t$ will be $T$(the period a.k.a. the time taken for one revolution/oscillation): $$\omega=\frac{2\pi}{T}$$

Because frequency, $f=\frac{1}{T}$: $$ \begin{equation}\begin{aligned} \omega&=\frac{2\pi}{T}\\ &=2\pi\times\color{green}\frac{1}{T}\\ \omega&=2\pi \color{green}f\\ \end{aligned}\end{equation} $$

Centripetal acceleration and force

This is the acceleration of the object towards the center of the circle. It is given by the formula: $$a={\omega}^2 r$$

Recall that $F=ma$. Therefore, centripetal force (the force required to keep the object moving in the circle) is: $$F=m{\omega}^2r$$


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