circular motion

24 July

Centrifuge Toy

Seng Kwang Tan 06 Motion in a Circle, 3D Printing 0 Comments

I designed this 3D teaching tool using Tinkercad and printed it out so that my colleague can use it to demonstrate the effect of a centrifuge.

As the toy is being spun, the ball bearings will appear to be thrown outwards. The centripetal forces that are meant to keep them in circular motion is made up of friction and any contact force due to the curvature of the base. If the rate of spin is sufficiently high, there will be insufficient contact force keeping the ball bearings in a circular path and hence, they will spiral outward and land into the cups found near the ends when the spinning stops.

Anyone can 3D-print this design as it had been uploaded into Thingiverse. This is my first original submission and can be found here. You will need 4 tiny balls of no more than 8 mm in diameter. The top is to be covered with a clear sheet of plastic cut-out after tracing the shape using a marker. The sheet can be stuck on the top using normal glue. This plastic cover serves to ensure the balls do not fly out if spun too fast.

A centrifuge toy to demonstrate the effects of rotational motion with insufficient centripetal force.
16 July

Non-Uniform Vertical Circular Motion

06 Motion in a Circle, Demonstrations, GeoGebra 0 Comments

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.

24 May

Angular velocity

06 Motion in a Circle, GeoGebra 0 Comments

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.

24 May

Angular displacement

06 Motion in a Circle, GeoGebra 0 Comments

This GeoGebra app shows the relationship s = rθ.

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.