A-Level Maths: Moments, Beams and Ladders
Moments turn equilibrium problems from one equation into two: a rigid body at rest needs both zero resultant force and zero resultant moment about every point. This material trains the moments problems that appear in the mechanics part of A level Mathematics, based on the DfE subject content on moments in simple static contexts, together with resolving forces and the friction model.
It begins with the moment of a force, including a force acting at an angle, where you must use the perpendicular distance to the line of action or the perpendicular component of the force. You then work with beams and planks resting on two supports or hanging from two strings: finding reactions and tensions, choosing the point to take moments about so that an unknown force disappears, and locating the centre of mass of a non-uniform rod from the reactions.
Tilting problems ask how far a person can walk along a beam before it is about to turn about one support, which means setting the reaction at the other support to zero while keeping the equilibrium equations. A rod hinged to a wall and held by a string shows why moments about the hinge are so useful: the hinge reaction, which has both horizontal and vertical components, drops out.
The final group of problems covers ladders resting on rough horizontal ground against a smooth vertical wall. You use the fact that a smooth wall can only push horizontally, resolve to link the wall reaction with friction at the foot, take moments about the foot, and apply to find the least coefficient of friction, the smallest safe angle or how far up the ladder a person can climb.
All values are given in each question, with and answers to 3 significant figures.
The material offers three formats. The quiz has 12 worked multiple-choice problems with explanations that show which point to take moments about and why the wrong options are wrong. The flashcards cover the definitions and methods: moment of a force, conditions for equilibrium, centre of mass, tilting, smooth and rough contacts and hinge reactions. The written work has 8 longer problems to set out by hand, including a full ladder analysis and a hinged rod with its hinge reaction.
- Calculate the moment of a force, including a force at an angle to a rod
- Use the two conditions for equilibrium of a rigid body
- Find reactions at supports and tensions in strings for loaded beams
- Locate the centre of mass of a non-uniform rod
- Solve problems where a beam is on the point of tilting
- Analyse a rod hinged to a wall, including the hinge reaction
- Solve ladder problems with a smooth wall and rough ground using F ≤ μR
Practice material written by Zestly, based on the DfE A level Mathematics subject content (mechanics: moments in simple static contexts, resolving forces, the F ≤ μR model of friction).
Sample question
A light rod AB is smoothly pivoted at A. A force of 30 N acts at B, where AB = 1.5 m, in a direction making an angle of $60^\circ$ with the rod. What is the magnitude of the moment of the force about A?
- $22.5$ N m
- $39.0$ N m
- $45.0$ N m
- $26.0$ N m
See the answer
$39.0$ N m
Moment $=$ force $\times$ perpendicular distance from A to the line of action. The perpendicular distance is $1.5\sin 60^\circ = 1.299$ m, so the moment is $30 \times 1.299 = 39.0$ N m (3 s.f.). Equivalently, use the component of the force perpendicular to the rod, $30\sin 60^\circ$, times 1.5 m. $22.5$ N m uses $\cos 60^\circ$ and $45.0$ N m ignores the angle.