Moments Levers and Gears

What is the meaning of moment in physics?

When we apply a force to an object in a certain direction, we can make it rotate. The point around which the rotation occurs is called the pivot point. Moments are all about understanding and quantifying the effectivity of a force to make an object turn.

To quantify the moment of a force,$M$,we multiply the applied force$F$, by the perpendicular distance of the line of action of the force from the pivot point,$d$.

$M=Fd$

$\text{moment of a force = force × distance}$

Since the units of force are newtons$\left(\mathrm{N}\right)$ and units of distance are metres$\left(\mathrm{m}\right)$, we measure moment in newton-metres$(\mathrm{N}\mathrm{m})$.

The following picture shows how to generate a moment in a nut using an adjustable wrench by applying a force in the correct direction. In this case, the centre of the nut is the pivot point.

The force applied to the wrench causes a moment that makes the nut turn around its centre, StudySmarter Originals

Causes of moments

From the definition provided previously, we know that force cause moment. But not just any force! For example, note that if the line of action of a force passes through the pivot, there will be no moment, as the perpendicular distance is zero. Hence, there is no rotation in such a case.

The line of action of the force passes through the pivot point. In this case, the wrench does not make the nut turn as there is no moment. StudySmarter Originals

Using this idea, we can balance objects so that they do not fall. We just need to make sure that their weight - which is directed downwards - is acting in a direction that passes through the point of support of the structure that we are trying to balance - the pivot. Do you remember the definition of the centre of mass?

Balancing an object is easy if we know where the centre of mass is. Consider the bottle shown at the beginning. If the bottle is inclined, its centre of mass does not align with the support point. Therefore, its weight will act at a certain perpendicular distance from the pivot, creating a moment. This moment makes the bottle turn towards the surface it is on, and it falls.

The weight of the bottle generates a moment because the centre of mass is not aligned to the pivot point. StudySmarter Originals

However, the slanted bottle holder is designed so that the centre of mass of the bottle holder-bottle system is directly above the support point. This way, the weight of the system does not generate a moment.

The centre of mass of a bottle holder-bottle system is aligned with the pivot point. Therefore, the system is not subject to any moment because of its weight. StudySmarter Originals

Principle of Moments

We now know that moments can cause an object to rotate, but as in the previous examples, sometimes we may prefer our system to remain static. Consider the structures lifted during construction. It is very important that these structures are not swinging around because it would be too dangerous. However, we can not always balance objects using their centre of mass. In cases like those, we need to balance the moments, and the principle of moments can help us with that.

An object will only rotate if there is an imbalance of moments in the clockwise and anti-clockwise directions. Therefore, we can prevent an undesired moment from causing a rotation clockwise by balancing it using an anti-clockwise moment of the same magnitude.

When a system is balanced, it is said to be in equilibrium. An example of this is when two people of the same mass sit on either side of a see-saw. Since their weights are equal, so are the distances from the pivot at the centre to either force. The generated moments are equal because of the equal forces - same weights - and perpendicular distances to the pivot. Therefore, the turning effects from the two forces cancel each other out, and the see-saw does not rotate. Note that there is also a reaction force at the pivot (represented as$F_p$in the diagram below). However, this does not cause any moment because the perpendicular distance for it is zero.

The moments due to either child on a see-saw depend on their masses and their distances from the pivot, philschatz

More examples and applications of moments: levers and gears

With the examples discussed, you might have already started to realize how often moments are present in everyday life. Some other applications of moments include scissors and cutting pliers, which we can use to cut even through metal with just our hands' strength. Another example is removing a nail using a hammer's head and pushing or pulling the hammer's grip. The example of the hammer and nail is illustrated below. Remember that we have to consider the perpendicular distance from the pivot to the line of action of the force, not the distance from the pivot to where the force is applied.

A hammer removing a nail is an application of moments, StudySmarter Originals

All these particular applications of moments are known as levers. Let's see how they work in more detail.

What is a lever and how does it work?

We can use levers to lift heavy weights more easily. Levers work on the principle that we can produce a greater moment (and thus a larger turning effect) by applying a smaller force at a greater distance from the pivot. The idea is for the fulcrum to be closer to the load - the weight that we want to lift - than it is from the point where we exert the force or effort. Have a look at the picture below. In this case, the load is a big rock. The rock's weight generates a moment on the lever system. However, we can create a moment equal or greater using less force because the distance where the effort is applied is greater.

A lever uses moments to lift a heavy weight. The load to be lifted causes a moment. However, we can cause a greater moment by applying less force than the weight of the load if we apply it at a larger distance from the fulcrum. StudySmarter Originals

A wheelbarrow is another example of the applications of levers. Loads that are too heavy to carry directly can be transported using wheelbarrows instead. The load is placed at the bottom above a wheel, and two handlebars come up it. These handlebars can be lifted using less force than the load itself as the effort is applied further from the pivot than the load is.

P represents the pulling force or 'effort' applied by the person, and R represents the load resistance due to its weight. The pivot is located at the centre of the wheel, tec.amordediocadiz

The diagram above shows the forces acting in this situation. Both forces act at the left of the pivot of the system, which is at the centre of the wheel. However, these forces are acting in opposite directions. The load's weight is causing an anti-clockwise moment. On the other hand, the force the person applies by pulling the handlebars - the effort - generates a clockwise moment.

We can see that levers are very useful, but they are not the only applications of moments. We also apply this concept when using gears.

What is a gear and how does it work?

You may have seen gears in various places, such as bicycles and pulleys. Gears, yet simple, can be very useful since they have different applications. For example, when gears are connected to each other, they can turn at different rates because for a bigger gear to complete a turn, a smaller gear needs to complete more turns. The exact amount of turns depends on the gears' sizes. Note that if one gear is spinning anti-clockwise, this causes the connected gear to rotate clockwise. Therefore, we can use them to change the direction in which an object is spinning as well.

If one gear turns in one direction, this will result in the other turning in the opposite direction, Wikimedia commons

But how exactly is the concept of moment applied here? At the point of contact for the gears' teeth, a force equal and opposite is exerted on each other. However, as their radii are different, the moments acting on either gear are also different. Applying a force to the smaller gear will cause a greater moment on the bigger gear because the force applied to it will be the same, but its radius is greater.