## What is a Moment of Force?

The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.

The moment of a force is the product of a force and its distance from a fixed point. Often the fixed point is the fulcrum or the axis of rotation or the pivot point. Assume a body pivoting about a point P. A force F acting on the body, acting at distance d from P, will generate a moment of M = F x d. The moment of force is a vector, whose magnitude equals the product of Force and the perpendicular distance from the axis.

In many practical examples, the moment of a force has a rotational effect on the system.

**An example of a moment of force**

The simple lever is a good system that works according to the principles of “moment of force”. A simple lever has a fulcrum or pivot, a load, and the point at which force is applied. About a fulcrum, the moment of a force serves to reverse the direction of application of the force.

In the illustration above, an object of mass M is on the left end of the lever. The fulcrum is at point F, the green shows the forces acting on the system and the red arrow depicts the weight of the load (M x g). When the system is in equilibrium, the forces acting about the fulcrum are equal. In other words, the force exerted on the right-hand side of the lever equals the weight of the object on the left. Moreover, in equilibrium, the moments about F are equal. If we only consider the forces acting to the left of F, the moment (left) is given by:

M_{L}=Mg x d_{1}

where d_{1} is the distance between the mass and F. At equilibrium, this is equal to the moment of the forces acting on the right side. Thus,

M_{L} = M_{R}

Mg x d_{1} = M_{R}

Assuming a force F_{R} acting at a distance of d_{2} from F, then M_{R} is given by:

M_{R} = F_{R} x d_{2}

It must be noted that the distances (d_{1} and d_{2}) are the perpendicular distances from the pivot.

Fixing d_{2} or knowing d_{2 }allows for an easy way to calculate the force required to achieve equilibrium.

F_{R} = M_{L} / d_{2}

F_{R}= Mx g x d_{1} / d_{2}

This can further be simplified by assuming a unit force on the left (M= 1). Then the force needed to lift the unit mass object situated d_{1} units away from the fulcrum is proportional to the ratio of distances from the fulcrum.

F_{R } ∝ d_{1} / d_{2}

The moment of force acting on a lever is calculated assuming that the force is perpendicular to the body/system on which they act. When the lever is at an angle to the perpendicular plane, then the effective length or the “perpendicular distance” is reduced depending on the angle θ as cos(θ) and hence reduces the effective moment. For a given force, then the generic formula for the effective moment is as follows:

M_{eff} = F * L * cos(θ)

Thus, when using a spanner, the turning effect of a downward or upward force is maximum when the spanner arm is perpendicular to the force applied. A steeper angle (approaching 90 degrees) means a proportionately smaller effective moment or turning effect that is available to cause rotation about the center.

## Moment of a Couple and Torque

A couple is a set of two opposing parallel forces, with non-coinciding lines of action. When a couple acts on an object connected to a pivot, it generates torque, which in turn causes rotation. Torque is the vector product of Force and distance that has the same units as the moment of force [Nm] and is perpendicular to the plane of the forces. Mathematically, it is the cross product of the Force vector and the perpendicular distance from the axis, or the distance separating the couple. Orientation and relative positioning of the forces in the couple will determine the direction of rotation, generating either clockwise moments or anticlockwise moments.

If the force applied to a body rotates the body in an anticlockwise (counterclockwise) direction, then, the moment of force is referred to as anticlockwise moment; this is taken to be positive. If the force applied to a body rotates the body in a clockwise direction, then, the moment of force is referred to as clockwise moment; this is taken to be negative.

Torque, denoted by τ, is also useful in determining the ability of a force to turn a certain moment of inertia given a radius. In electric motors and engines, it denotes the ability to rotate a mass given the distance from the axis of rotation.

The direction of the moment of force or torque can be determined using the right-hand rule. When the fingers of the right hand are curled around the sense of rotation caused by two forces, then, the thumb indicates the direction of the moment of force.

## Bending and Shearing

The turning effect of forces, whether on levers or wheels (turning about an axis) are common examples of the moment of force. However, other effects are present ubiquitously but not commonly studied. These involve bending and shearing.

When an object is unable to rotate about an axis, it undergoes either bending or shearing. Of the two, bending is easily observed. For small deformities, such forces will create a vibration that resembles a simple harmonic motion in the structure.

## Context and Applications

This topic is studied in

- Bachelors in Science (Physics)
- Masters in Science (Physics)

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