06 Motion in a Circle

3D printed teaching aids

I bought a Creality Ender 3D printer in 2020 (going at about $270 at Lazada now), at the height of the pandemic and have been using it to print physics-related teaching aids for a while, including balloon hovercrafts, catapults, a Pythagorean cup, tippy top and a vertical axis wind turbine. In addition to complete demonstration sets, it is also handy for printing parts to fix old demonstration sets such as a base for a standing cylinder with spouts at different heights.

The Creality Ender 3 3D printer

This is a video compiled with the objects that I printed in recent months. The lime green filament that I used were purchased at $16.40 for 1 kg from Shopee. Therefore, each of the prints shown in the picture cost between forty cents to four dollars’ worth of filament.

The first is a coin funnel that can be used to demonstrate how centripetal force keeps objects moving in circles. As the energy of the coins decreases due to friction, the radius of the circle gets smaller and its speed actually increases. This forms a cognitive dissonance that often surfaces when we discuss satellites losing altitude in orbit.

The second is a tensegrity structure which can be used to teach about moments and equilibrium.

The third is a marble run set that was really just lots of fun to watch rather than teaching any difficult concept other than energy changes.

The fourth is a series of optical illusions that can be used to promote thinking about how light from reflections travel.

The final print is a cup holder that can be swung in vertical loops with a cup full of water. This is the most popular print among my colleagues and will certainly be used in term 3 for the JC1 lessons on circular motion.

Non-Uniform Vertical Circular Motion

Using a chain of rubber bands, I swung a ball around in a vertical loop. This demonstration shows how the tension in an elastic band changes according to the position of the ball, by referring to the length of the elastic band.

Securing the elastic band to the ball with a shoelace

When the ball of mass $m$ is at the bottom of the loop, the centripetal force is given by the difference between tension $T_{bottom}$ and weight $mg$, where $T_{bottom}$ varies depending on the speed of the ball $v_{bottom}$ and the radius of the curvature $r_{bottom}$.

$T_{bottom} – mg = \dfrac{mv_{bottom}^2}{r_{bottom}}$

When the ball is at the top of the path, it is given by

$T_{top} + mg = \dfrac{mv_{top}^2}{r_{top}}$

As the weight is acting in the same direction to tension when the ball is at the top, a smaller tension is exerted by the elastic band to maintain a centripetal force. Therefore , $T_{bottom} > T_{top}$.

The GeoGebra app below shows a simpler version of a vertical loop – a circular path with a fixed radius $r$. Consider a ball sliding around a smooth circular loop. The normal contact force varies such that

$N_{bottom} = \dfrac{mv_{bottom}^2}{r} + mg$

$N_{top} = \dfrac{mv_{top}^2}{r} – mg$

It can be shown that the minimum height at which the ball must be released in order for it to complete the loop without losing contact with the track is 2.5 times the radius of the frictionless circular track.

If we were to consider the rotational kinetic energy required for the ball to roll, the required initial height will have to be 2.7 times the radius, as shown in the video below:

Many thanks to Dr Darren Tan for his input. Do check out his EJSS simulation of a mass-spring motion in a vertical plane, which comes with many more features such as the ability to vary the initial velocity of the mass, graphs showing the variation of energy and velocity, as well as an option for a mass-string motion as well.

Centripetal force on and off

Here’s a quick video to demonstrate the movement of a ball initially moving in a circle before its centripetal force (contact force by the circular wall) is removed. The ball is observed to move in a tangent to the circle, in accordance with Newton’s 1st Law, since there is no longer a net force acting on it.

Angular velocity

This GeoGebra app shows how angular velocity ω is the rate of change of angular displacement (i.e. $\omega=\dfrac{\theta}{t}$) and is dependent on the speed and radius of the object in circular motion (i.e. $v=r\omega$).

Students can explore the relationships by doing the following:

Keeping r constant and varying ω.

Keeping ω constant and varying r.

Keeping v constant by varying r and ω.

Angular displacement

This GeoGebra app shows the relationship s = .

One activity I get students can do is to look at the value of θ when the arc length s is equal to the radius r. This would give the definition of the radian, which is the angle subtended at the centre of a circle by an arc equal in length to its radius.

Mathematics defines the constant π as the ratio of a circle’s circumference to its diameter. This can also be shown in the app, although you need to drag the moving point to a point just short of one full revolution.

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.

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