Moments of a forceEdexcel International A Level Maths: Revision notes

Section 1

The moment of a force

The moment of a force about a point measures its turning effect. It is the force multiplied by the perpendicular distance from the point to the line of action of the force: $\text{moment}=F\times d.$ The unit is the newton metre (N m). A moment is clockwise or anticlockwise; choose one sense as positive. A force of 15 N acting at 2 m from $A$ perpendicular to a rod has moment $15\times2=30$ N m about $A$ .

Key termsmomentline of actionperpendicular distance

Common mistake

Using the distance along the rod when the force is not perpendicular to it. The distance must be perpendicular to the line of action.

Section 2

Conditions for equilibrium of a rod

A rigid body (such as a rod or plank) under coplanar parallel forces is in equilibrium when two conditions hold:

the resultant force is zero (upward forces equal downward forces)
the total moment about any point is zero (the sum of clockwise moments equals the sum of anticlockwise moments, the principle of moments). Choosing the point about which to take moments carefully removes an unknown force from the equation, because a force through that point has no moment about it.

Key termsequilibriumprinciple of moments

Exam tip

Take moments about the point where the unknown force you do not need acts. Then use vertical resolving to find the other force.

Section 3

Rods, planks and beams

A uniform rod has its weight acting at its midpoint (centre of mass). A light rod has negligible weight. Weight is $mg$ with $g=9.8$ m s $^{-2}$ . A person or particle on the rod is treated as a point load at their position. Supports apply reactions (upward forces) at the points of contact; strings apply tensions. Draw a diagram marking every force with its distance from a chosen point.

Key termsuniformlightreactioncentre of mass

Common mistake

Forgetting the weight of a uniform rod, or putting it at an end instead of the midpoint.

Section 4

Worked example: two supports

A uniform plank $AB$ , 4 m long and of mass 30 kg, rests on supports at $A$ and $B$ . A man of mass 60 kg stands 1 m from $A$ . Moments about $A$ : $4R_B=30g\times2+60g\times1=1176$ , so $R_B=294$ N. Vertically: $R_A+R_B=90g=882$ , so $R_A=588$ N. Check: moments about $B$ give $4R_A=30g\times2+60g\times3=2352$ , $R_A=588$ N, as required.

Exam tip

Check your answer by taking moments about a different point.

Section 5

Tilting and limiting equilibrium

If a rod rests on two supports and a load moves towards one support, the reaction at the other support decreases. When that reaction becomes zero the rod is about to tilt about the first support. Set $R=0$ for that support and take moments about the one that remains. Example: a plank (weight at 1.5 m from $D$ ) is about to tilt about $D$ when a woman of weight $50g$ stands at distance $d$ from $D$ on the other side. $25g\times1.5=50g\times d$ gives $d=0.75$ m. A string with zero tension has the same effect: set its tension to zero.

Key termstiltingabout to tilt

Common mistake

Saying the rod tilts when a reaction is small. It is on the point of tilting when the reaction is exactly zero.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Moments of a force

A light rod $AB$ of length 2 m is horizontal. A vertical force of 15 N acts downwards at $B$ , and a vertical force of 24 N acts upwards at the point $C$ on the rod, where $AC=0.5$ m.
The 24 N force is moved to a point $D$ on the rod, so that the total moment of the two forces about $A$ is zero. Find $AD$ .2 marks
A uniform plank $AB$ of length 4 m and mass 30 kg rests horizontally on two supports, one at $A$ and one at $B$ . Take $g=9.8$ m s $^{-2}$ .
With the man still standing in that position, find the reaction of the support at $A$ .2 marks
A uniform rod $AB$ of length 3 m and mass 8 kg is held in a horizontal position by two vertical light strings, one attached at $A$ and the other attached at the point $C$ on the rod, where $BC=1$ m. Take $g=9.8$ m s $^{-2}$ .
Find the tension in the string at $A$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).