U3AOS1 Topic 3: Circular Motion

The distance around a circle is the circumference and is calculated through the following formula

\[ \displaystyle \Large d = \text{circumference} = 2 \pi r \]

$$d=circumference=2πrd = circumference = 2\pi r

Therefore the average speed can be calculated by dividing the distance around the circle by the time it takes

\[ \displaystyle \Large \text{avg speed} = \frac{d}{t}= \frac{2 \pi r}{t}\]

$avg speed =\frac{d}{t} = \frac{2\pi r}{t}$

The average velocity will always be 0 since the object returns to its original position (no displacement). However, there is a varying instantaneous velocity since the direction is changing.

The acceleration is calculated through the following formula

\[ \displaystyle \Large a = \frac{v^2}{r}= \frac{4 \pi^2 r }{T}\]

$a =\frac{v^2}{r} = \frac{4\pi^2 r}{T}$

The acceleration always points toward the center of the circle. So the net force is always toward the center of the circle as well.

Using these equations, the force on a mass travelling in circular motion can be calculated by using Newton's Laws.

\[ \displaystyle \Large F_{\text{net}} = ma = \frac{mv^2}{r}= \frac{4m \pi^2r }{T}\]

$F_{net} = ma= \frac{mv^2}{r} = \frac{4m\pi^2r}{T}$

A key note about circular motion is that the acceleration and the velocity are 90 degrees out of phase. This means that if the velocity at \( t=0 \)  $\mathrm{<span\; style="font-size:\; 12px;">t</span>}<span\; style="font-size:\; 12px;">=</span>\mathrm{<span\; style="font-size:\; 12px;">0</span>}$t =0is to the right then the acceleration would be either up or down depending on the direction it is travelling in. This can be seen in the simulation below.