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Resultant moments

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Summary

Resultant moments

In a nutshell

When there are a number of forces acting on a body, the turning effect of the resultant moments of the forces can be worked out. When a force $F$ acts perpendicular from a pivot at a distance $d$ , you can work out the moment $M$ using $M=F\times d$ . When a force is not perpendicular to the pivot and is instead at an angle of $\theta$ , you can work out the moment using trigonometry: $M=F\times d\sin\theta$ .

Variable definitions

QUANTITY NAME	SYMBOL	UNIT	UNIT SYMBOL
$Moment$	$M$	$Newton \space metre$	$Nm$
$Force$	$F$	$Newton$	$N$
$Distance$	$d$	$Metre$	$m$

Equations

DESCRIPTION	EQUATION
Moment of $F$ about a pivot, where $F$ and $d$ are perpendicular.	$M = F \times d$
Moment of $F$ about a pivot, where $F$ and $d$ are not perpendicular, but make an acute angle of $\theta$ .	$M = F \times d \sin\theta$

Note: These formulae are the same, since when $F$ and $d$ are perpendicular, $\theta=90$ and hence $\sin(\theta)=1$ .

Sum of moments

The resultant moment is the sum of the moments acting on a body. To determine the turning effect around a point, you choose to take clockwise or anticlockwise as the positive direction and then find the sum of the moments produced by each force.

Example 1

The diagram shows two forces acting on a lamina. Calculate the resultant moment about the point $P$ .

Maths; Moments; KS5 Year 13; Resultant moments

Find the moment of each force:

Moment of $10N$ force:

$\begin{aligned}=10 &\times 2\sin(90)\\ =10 &\times 2\\ = 20&Nm\ anticlockwise\end{aligned}$ 

Moment of $8N$ force:

$\begin{aligned}=8 &\times 1.5\sin(90)\\ =8 &\times 1.5\\ = 1&2Nm\ Clockwise\end{aligned}$ 

By taking the anticlockwise as the positive direction, the resultant moment is

$=20 + (-12) = 8$ 

The resultant moment is $\underline{8Nm}$ anticlockwise.

Note: $8Nm$ anticlockwise is the same as $-8Nm$ clockwise.

Example 2

This diagram shows some forces acting on a light rod. Calculate the resultant moment about the point $P$ .

Take clockwise as the positive direction.

Moment of $7N$ force:

$\begin{aligned}&=7\times 11\\ &= 77Nm\ clockwise\end{aligned}$ 

Moment of $3N$ force:

$\begin{aligned}&= 3\times 3\\ &=9Nm\ anticlockwise\end{aligned}$ 

Moment of $1N$ force:

$\begin{aligned}&=1\times 3\\ &=3Nm\ anticlockwise\end{aligned}$ 

Resultant moment:

$= 77 + (-9) + (-3) = 65$ 

The resultant moment is $\underline{65Nm}$ clockwise.

Example 3

This diagram shows forces acting on a light rod. Calculate the resultant moment about the point $P$ .

Take clockwise as the positive direction.

Moment of $5N$ force:

$\begin{aligned}&=5\times 4\sin(80)\\ &=20\sin(80)\\&\approx 19.7Nm\ clockwise\end{aligned}$ 

Moment of $6N$ force:

$\begin{aligned}&=6\times 3\\ &= 18Nm\ anticlockwise\end{aligned}$ 

Moment of $4N$ force:

$\begin{aligned}&=4\times 3\sin(40)\\ &=12\sin(40)\\ &\approx7.7Nm\ anticlockwise\end{aligned}$ 

Resultant moment:

$=20\sin(80) + (-18) + (-12\sin(40)) \approx -6.02$ 

The resultant moment is approximately $\underline{6Nm}$ anticlockwise.

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The principle of moments

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