# 06 Motion in a Circle

## Uniform vertical circular motion

The following GeoGebra app simulates the force vectors on an object in uniform vertical circular motion.

A real world example of this would be the forces acting on a cabin in a ferris wheel.

<iframe scrolling="no" title="Vertical Uniform Circular Motion " src="https://www.geogebra.org/material/iframe/id/t5jstqsm/width/640/height/480/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/false/ctl/false" width="640px" height="480px" style="border:0px;"> </iframe>

## Vertical Non-Uniform Circular Motion

This is a simulation that shows the vectors of forces acting on an object rolling in a vertical loop, assuming negligible friction.

To complete the loop, the initial velocity must be sufficiently high so that contact between the object and the track is maintained. When the contact force between the object and its looping track no longer exists, the object will drop from the loop.

The following code is for embedding in SLS.

<iframe scrolling="no" title="Vertical non-uniform circular motion" src="https://www.geogebra.org/material/iframe/id/ny3jhhsp/width/640/height/480/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/false/ctl/false" width="640px" height="480px" style="border:0px;"> </iframe>

## Aircraft Turning in a Circle: a 3-D Visualisation with GeoGebra

This GeoGebra app is a 3-D visualisation tool of the force vectors acting on an aircraft turning with uniform circular motion in a horizontal plane.

I prepared this in advance as I will be lecturing on this JC1 topic next year.

## Cardboard Boomerang

A indoor boomerang can be constructed using 3 strips of cardboard put together. Throwing it may require some practice though but when you get the hang of it, it can inject great fun into your lesson. You can explore using different types of material to get the best boomerang.

Materials

1. Cardboard about 1 mm thick, of suitable rigidity
2. Staples
3. Scissors
4. Rubber band or tape for added weight

Procedure

1. Cut 3 equal rectangular strips of cardboard measuring 12 cm x 2.5 cm. You may like to trim the sharp corners on one of the ends of each strip.
2. Cut a slit of 1.5 cm along the middle of each strip, on the untrimmed end.
3. Join the strips together at the slits, the angle between two adjacent strips being 120 degrees.

4. One side of the slit should overlap another so that it looks like the above:
5. Staple the overlapping centre together.
6. The boomerang is ready for use! Throwing the boomerang is done by holding onto one of the wings. The boomerang should be almost vertical, at an angle of about 10o. With a flick of the wrist, spin the boomerang as it leaves the hand. The direction of spin should be toward the side that is tilted up.

Science Explained
A boomerang requires a centripetal force to cause it to fly in a circular path back to the thrower. This centripetal force comes from the lift that the wings generate as they cut through the air.

## Angular Displacement – 2011 A-level question

A disc rotates clockwise about its centre O until point P has moved to point Q, such that OP equals the length of the straight line PQ. What is the angular displacement of OQ relative to OP?

A.   $\frac{\pi}{3}$ rad

B.   $\frac{2\pi}{3}$ rad

C.   $\frac{4\pi}{3}$ rad

D.   $\frac{5\pi}{3}$ rad

The triangle OPQ is equilateral, so the angle $\angle QOP$ = 60° or $\dfrac{2\pi}{6}=\dfrac{\pi}{3}$ rad.
As OQ is displaced clockwise from OP, angular displacement $\theta = 2\pi – \dfrac{\pi}{3} = \dfrac{5\pi}{3}$ rad.