Moment Of A Force

The moment of a force or torque, r is defined as the turning effect of the force about a pivot and is the product of the force (F) and the perpendicular distance (d) from the line of action of the force to the pivot.

SI unit of moment of a force is Newton-metre (Nm). It is a vector quantity.
Its direction is given by the right-hand grip rule perpendicular to the plane of the force and pivot point which is parallel to the axis of rotation.

$$r = F \times d$$

,where
r is the moment of force/torque
F is the force
d is the perpendicular distance from the line of action of the force to the pivot

Couple

A couple is a pair of forces, equal in magnitude but opposite in direction, whose lines of motion do not coincide.

Will still rotate as there is a net moment
As forces are equal and opposite, resultant force is zero and so there is no linear acceleration
$r = F \times d$

Torque(Moment of a couple) is the product of one of the forces and the perpendicular distance between their lines of action of the forces.

Can take moment from any point

Self-Test Questions

Consider the figure above. Given that the cat weighs 150 N and the distance from the cat to the pivot is 30 cm, calculate the moment of force about the pivot.

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$$\begin{aligned} r &= F \times d \\ &= 150 \times 0.30 \\ &= 45 \text{ Nm} \end{aligned}$$

Why is it easier to open the lid of a container with a spoon than a coin?

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A spoon is longer than a coin. If you use them to open the lid of a container by pivoting, the perpendicular distance (d) will be longer for a spoon than a coin.

Let’s assume that it takes an $x \text{ Nm}$ of moment of a force to open the lid and that $d_{\text{spoon}} = 2 d_{\text{coin}}$.

Find amount of force needed to open the lid if we use a spoon:

$$\begin{aligned} r &= F \times d \\ x &= F_{\text{spoon}} \times d_{\text{spoon}} \\ F_{\text{spoon}} &= \frac{x}{d_{\text{spoon}}} \end{aligned}$$

Find amount of force needed to open the lid if we use a coin:

$$\begin{aligned} r &= F \times d \\ x &= F_{\text{coin}} \times d_{\text{coin}} \\ F_{\text{coin}} &= \frac{x}{d_{\text{coin}}} \end{aligned}$$

Since $d_{\text{spoon}} = 2 d_{\text{coin}}$,

$$\begin{aligned} F_{\text{coin}} &= \frac{x}{\frac{d_{\text{spoon}}}{2}} \\ &= 2 \frac{x}{d_{\text{spoon}}} \\ &= 2 F_{\text{spoon}} \end{aligned}$$

From the calculations above ($F_{\text{coin}} = 2 F_{\text{spoon}}$), you will need twice as much force to open the lid of a container if you use a coin.