When applying the principle of moments to calculate the magnitude of a force creating a turning effect, where the force is not perpendicular to the length of the object, there are two approaches.

Take the following problem:

A uniform rectangular beam has negligible thickness and weight 850 N. Its length is 5.0 m and it is in contact with the top of a support at point P. P is 0.80 m from one end of the beam.

The beam is held stationary, at an angle of 30° to the horizontal, by a rope that is attached to the bottom corner of the other end of the beam.

Calculate the magnitude of the force T on the beam due to the tension in the rope.

Approach 1: Identifying the perpendicular distance between line of action of the force and pivot

In the applet above, check the box that says “Show perp dist” to see the lines representing the perpendicular distances between each line of action of the force and the pivot P.

Taking moments about P,

Clockwise moment due to T = Anti-clockwise moment due to W

There is a new internet trend called “tensegrity” – an amalgamation of the words tension and integrity. It is basically a trend of videos showing how objects appear to float above a structure while experiencing tensions that appear to pull parts of the floating object downwards.

In the diagram below, the red vectors show the tensions acting on the “floating” object while the green vector shows the weight of the object.

The main force that makes this possible is the upward tension (shown below) exerted by the string from which the lowest point of the object is suspended. The other tensions are downward and serve to balance the moment created by the weight of the object. The centre of gravity of the “floating” structure lies just in front of the supporting string. The two smaller downward vectors at the back due to the strings balance the moment due to the weight, and give the structure stability sideways.

This is a fun demonstration to teach the principle of moments, and concepts of equilibrium.

The next image labels the forces acting on the upper structure. Notice that the centre of gravity lies somewhere in empty space due to its shape.

These tensegrity structures are very easy to build if you understand the physics behind them. Some tips on building such structures:

Make the two strings exerting the downward tensions are easy to adjust by using technic pins to stick them into bricks with holes. You can simply pull to release more string in order to achieve the right balance.

The two strings should be sufficiently far apart to prevent the floating structure from tilting too easily to the side.

The centre of gravity of the floating structure must be in front of the string exerting the upward tension.

The base must be wide enough to provide some stability so that the whole structure does not topple.

Here’s another tensegrity structure that I built: this time, with a Lego construction theme.

Apart from using Lego, I have also 3D-printed a tensegrity structure that only requires rubber bands to hold up. In this case, the centre of gravity of the upper structure is somewhere more central with respect to the base structure. Hence, 3 rubber bands of almost equal tension will be used to provide the balance. The STL file for the 3D model can be downloaded from Thingiverse.com.

The main challenge in assembling a tensegrity structure is the adjustment of the tensions such that the upper structure is balanced. One way to simplify that, for beginners, is to use one that is supported by rubber bands as the rubber bands can adjust their lengths according to the tensions required.

Another tip is to use some blu-tack instead of tying the knots dead such as in the photo below. This is a 3D printed structure, also from Thingiverse.

(This post was first published on 18 April 2020 and is revised on 24 August 2022.)

This GeoGebra interactive allows students to vary the position of the centre of gravity of a shelf in order to observe the changes of the other two force vectors. The position of the supporting cable can be adjusted too.

The ability to resolve vectors allows students to apply principle of moments to understand how the vertical components of each force vary.

This is meant for the JC1 topic of Forces.

To embed into SLS, you can use the following code:

<iframe scrolling="no" title="Equilibrium of a Wall Shelf" src="https://www.geogebra.org/material/iframe/id/xdbr7qr5/width/700/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" width="700px" height="500px" style="border:0px;"> </iframe>

While preparing for a bridging class for those JAE JC1s who did not do pure physics in O-levels, I prepared an app on using a vector triangle to “solve problems for a static point mass under the action of 3 forces for 2-dimensional cases”.

For A-level students, they can be encouraged to use either the sine rule or the cosine rule to solve for magnitudes of forces instead of scale drawing, which is often unreliable.

For students who are not familiar with these rules, here is a simple summary:

Sine Rule

If you are trying to find the length of a side while knowing only two angles and one side, use sine rule:

$$\dfrac{A}{\sin{a}}=\dfrac{B}{\sin{b}}$$

Cosine Rule

If you are trying to find the length of a side while knowing only one angle and two sides, use cosine rule:

The following GeoGebra app simulates a pressure sensor that measures hydrostatic pressure, calibrated to eliminate the value of atmospheric pressure.

The purpose of this simulation is to address certain misconceptions by students such as the assumption that the shape of a container affects the pressure such that the pressure differs in different containers when measured at the same depth.

Drag the dot around to compare the pressure values at the same height between both containers.

The following codes can be used to embed this into SLS.

This simple demonstration can be done anywhere at home using the following items:

an empty glass

a toothpick

two forks

The video below demonstrates how to do it. When the forks are balanced on the mouth of the glass with the toothpick, the centre of gravity of the forks-and-toothpick system will adjust itself so that it lies vertically below the pivoting point. This is possible because the forks form a V-shape within which the centre of gravity can exist.