Moments and Equilibrium

Moment and equilibrium relationship

If you take your little finger and try to shut an open door with it by placing your finger near the door handle, the door would close easily. Now try to close the door with your little finger again, but this time try to push the door by placing your finger near the hinge. Does the door close as easily as when you placed your little finger near the handle?

It would be much more difficult to close the door with your finger placed near the hinge. Why is that? The distance, or specifically the perpendicular distance, from the hinge to the door handle is more, which is why the force generated to produce the turning effect is more as well.

Fig. 2: The distance \(d_2\) would generate a greater turning force as compared to \(d_1\).

We often use forces to cause an object to rotate, in other words, we need forces to create a moment. A moment can also explain why it is easier to use a longer spanner than a shorter spanner.

The force at point A would generate a smaller turning effect as compared to the force at point B. Wikimedia

Moment of a force

We can calculate the moment of a force using the equation:

\[M=Fd\]

Where $\mathrm{M}$ is the moment in Nm, $\mathrm{F}$ is the force in Newtons, and $\mathrm{d}$ is the distance along the normal to the force in m. One thing important to keep in mind in the above equation is that the distance $\mathrm{d}$ is the perpendicular distance of the line of action of the force to the pivot. So, a bigger force needs to be applied far away from the pivot point for a greater moment.

Perpendicular distance means a distance at right angles. In the figure below, the perpendicular distance is from the point at which the applied force acts A to the pivot B (centre of the nut) as the force and the distance to the pivot make right angles to each other.

The force applied makes a right angle to the distance from the pivot. Adapted from Wikimedia.

If the force applied happens not to be at right angles, as shown in the figure below, the moment produced in turn will also be smaller. The distance from the pivot B to the applied force A is not perpendicular in the figure below; but there is no need to worry, as the perpendicular distance will be given in any problem you will be asked to solve.

The force applied at an angle to the distance from the pivot does not count as the perpendicular distance. Adapted from Wikimedia.

Notice that a moment can be in a anticlockwise direction (positive direction) or a clockwise direction (negative direction) as it is essentially a rotation that can be in any direction depending upon the distance and the direction of the applied force. Now, of course, there can be more than one moment acting on an object as well, for example, two people on a seesaw. But, the clockwise and anticlockwise directions and multiple moments lead to an important concept in physics known as the principle of equilibrium of a moment or simply the principle of the moment.

The moment and equilibrium principle

A mechanical measuring scale like the one in the figure below demonstrates the moment and equilibrium principle. We use scales to compare the weight of the two bodies.

Scales are used to compare the difference in mass between two objects. Wikimedia.

When the weights are imbalanced, the instrument tilts such that the scale with the heavier object will lower and the scale with the lighter object will rise. This 'tilting' is the anticlockwise or clockwise moment produced because of the imbalance of downward forces on both sides. Because the length of each arm which the scales are suspended from are the same length, If we want to balance the instrument, the weight of objects on both sides needs to be the same. This will lead to the clockwise moment and anticlockwise moment becoming equal and hence, the body will be in equilibrium.

A balance is in equilibrium if the clockwise moment and anticlockwise moment become equal. Adapted from Wikimedia.

Moments and equilibrium examples

Let's say John is sitting on the right-hand side of the seesaw relative to us. The Force due to his weight is $600\mathrm{N}$ and he is sitting at a $2\mathrm{m}$ distance from the pivot. As mentioned previously, moments can be clockwise or anticlockwise and in John's case, the moment produced would be clockwise. The moment in the clockwise direction for John's case would be:

$$\begin{gathered} \mathrm{M}=\mathrm{Fd} \\ \mathrm{M}=600\mathrm{N}\times 2\mathrm{m} \\ \mathrm{M}=1200\mathrm{N}\mathrm{m} \end{gathered}$$

In this diagram, the total moment acting on the seesaw is clockwise (negative) because there is one moment acting clockwise due to the weight of the person sitting on the seesaw.

Soon afterward, Peter joins John on the seesaw which would create an anticlockwise moment. If we are to calculate the total moment of the seesaw, we would need to take into account both the clockwise moment and the anticlockwise moment and then see which moment is bigger than the other.

A clockwise and anticlockwise moment is generated by the seesaw as a result of two forces acting in the same direction on opposite ends. Adapted from Wikimedia.

If somehow the two moments become equal, meaning that the anticlockwise moment is equal to the clockwise moment, then the seesaw would not turn. In other words, if the moments become equal, then the seesaw would be in equilibrium, which is what the principle of the moment states.

Moments and equilibrium equation

In the scenario of Peter and John, what is the distance from the equilibrium Peter would need to sit for the seesaw to reach equilibrium? We calculated the moment generated by the force by John, so Peter's moment needs to be as same as John's moment. We know the force on Peter's end is $800\mathrm{N}$ so we can write,

$\begin{array}{rcl}\mathrm{Sum}\mathrm{of}\mathrm{clockwise}\mathrm{moments}& =& \mathrm{Sum}\mathrm{of}\mathrm{anticlockwise}\mathrm{moments}\\ {\mathrm{F}}_1{\mathrm{d}}_1& =& {\mathrm{F}}_2{\mathrm{d}}_2\\ 600\mathrm{N}\times 2\mathrm{m}& =& 800\mathrm{N}\times {\mathrm{d}}_2\\ {\mathrm{d}}_2& =& \frac{600\mathrm{N}\times 2\mathrm{m}}{800\mathrm{N}}\\ {\mathrm{d}}_2& =& 1.5\mathrm{m}\end{array}$

So Peter would have to sit at a $1.5\mathrm{m}$ distance from the pivot for the seesaw to be balanced.

Moments and equilibrium applications

There are a number of applications of moments and equilibria that we use in our day-to-day lives. Here we will list and discuss a few examples as a fun way to round off this article.

A classic example that we have already considered is the seesaw. This simple but fun piece of playground equipment wouldn't be able to operate without the principles of moments. The moment of on each end of a seesaw is supplied by the kicking action of the people sitting on either arm of the seesaw, or by them leaning back in order to change their centre of mass so that it is further from the pivot.

Another application we have studied is the example of weighing scales. Weighing scales operate under the principle that balanced clockwise and anticlockwise moments result in the system under consideration being in a state of equilibrium, i.e. there is no net movement because the net moment is equal to zero. We can balance an object on one side of the scales and incrementally add weights of known mass onto the other side of the scales until the weighing scales are balanced. Adding up the mass of the weights needed to balance the scales will yield the total mass of the object being weighed.