Circular Motion

Warning

If you are in year 4 (or year 3 if you take PO), you probably already know this

Also many of these are actually vector quantities

Introduction

When anything moves in a circle, it has an acceleration. This is because its velocity is constantly changing. Even if it's speed (magnitude of velocity) is constant, the direction is constantly changing.

For an object to go into a circle, at every point on the circle the following is true:

\[ a_c = \frac{v^2}{r}\]

where \(v\) is the velocity which is always tangent to the circle, and where \(a_c\) is the centripetal acceleration. This acceleration is always perpendicular to the direction of the velocity of the object, pointing towards the center of the circle.

Angular quantities

For many quantities in normal (linear, straight-line) mechanics, there exists a similar one for rational motion

linear quantity	angular equivalent	formula
displacement \(s\)	angular displacement \(\theta\)	\(s=r\theta\)
velocity \(v\)	angular velocity \(\omega\)	\(v=r\omega\)
acceleration \(a\)	angular acceleration \(\alpha\)	\(a=r\alpha\)

Note

for the following quantities instead of meters, angular quantities use radians. (\(\theta\) uses radians, \(\omega\) uses rad/s, etc.)

you can use your kinematics equations to calculate angular quantities by simply replacing every linear component with the angular counterpart, though usually there will not be any angular acceleration:

straight-line equation	angular equivalent
assuming \(a\) is constant	assuming \(\alpha\) is constant
\(v_f = v_i+at\)	\(\omega_f = \omega_i+\alpha t\)
\(s=v_it+\frac{1}{2}at^2\)	\(\theta=\omega_it+\frac{1}{2}\alpha t^2\)
\(v_f^2=v_i^2+2as\)	\(\omega_f^2=\omega_i^2+2\alpha\theta\)
\(s = \frac{1}{2}(v_i+v_f)t\)	\(\theta = \frac{1}{2}(\omega_i+\omega_f)t\)

Uniform Circular Motion (UCM)

This is a special case of circular motion, where speed is constant. Put differently, the only acceleration is the centripetal acceleration.

since the magnitude of \(v\) is constant, the magnitude of \(a_c\) is constant.

Centripetal force

since \(F = ma\), we can say the centripetal force \(F_c\) is:

\[\begin{align*}F_c &= \frac{mv^2}{r} \\ &= \frac{m}{r}(r\omega))^2 \\ &= mr\omega^2\end{align*}\]

One useful thing about is that the net force on an object underdoing UCM has to be equal to \(F_c\).

Frequency and Period

The period \(T\) is the amount of time it takes for the object to go around the circle. Now full circle is \(2\pi\) radians, so we can calculate the angular velocity \(\omega\) from \(T\).

\[\omega = \frac{2\pi}{T}\]

The frequency \(f\) is the amount of rotations the object goes throught in 1 second. It is also defined as \(f=\frac{1}{T}\). Therefore:

\[\omega = 2\pi f\]