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

Equations 

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

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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 $$P$$.

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.