Mechanics: Moments
Moments is the A Level Mechanics topic covering the turning effect of a force about a pivot, calculated as force multiplied by the perpendicular distance from the pivot to the line of action of the force. It is used to analyse rigid bodies such as beams, rods and see-saws held in equilibrium, usually alongside resolving forces vertically and horizontally. It builds directly on resolving forces and Newton's laws.
Before you start
Make sure you're comfortable with these topics first:
Method
- Identify the pivot, or a sensible point to take moments about, and list every force acting on the rigid body, including weight, reactions and applied forces.
- For each force, find its perpendicular distance from the pivot; if the force acts at an angle to the body, use force x perpendicular distance, found from the sine of the angle.
- Take moments about the pivot, treating clockwise and anticlockwise turning effects as opposite, and set the sum of clockwise moments equal to the sum of anticlockwise moments for equilibrium.
- Use vertical (and horizontal, if needed) equilibrium equations alongside the moments equation to find any remaining unknown forces.
- Choose your pivot strategically: taking moments about a point where an unknown force acts removes that force from the equation and simplifies the algebra.
- State the final answer with the correct units, newtons for forces and newton metres for moments, and check the result is sensible.
Worked example
A uniform beam AB has length 8 m and weight 240 N. The beam rests horizontally in equilibrium on two smooth supports, one at C, 1 m from A, and one at D, 2 m from B. Find the reaction at each support.
- Draw a diagram: the weight of 240 N acts at the midpoint of AB, 4 m from A; the reactions R_C (at C, 1 m from A) and R_D (at D, 6 m from A, since D is 2 m from B) act upward.
- Take moments about C to eliminate R_C: R_D x (6 - 1) = 240 x (4 - 1), i.e. R_D x 5 = 240 x 3.
- Simplify: R_D x 5 = 720.
- Solve: R_D = 144 N.
- Use vertical equilibrium: R_C + R_D = 240, so R_C = 240 - 144 = 96 N.
- Final answer: R_C = 96 N, R_D = 144 N.
Practice questions
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Q1A spanner is used to turn a bolt. A force of 40 N is applied at right angles to the spanner, 0.3 m from the bolt. Find the moment of the force about the bolt.Show answer
Answer: 12 N m (40 x 0.3)
Q2A force of 25 N acts at a perpendicular distance of 0.6 m from a pivot. Find the moment of the force about the pivot.Show answer
Answer: 15 N m (25 x 0.6)
Q3A force of 50 N is applied to a lever at an angle of 60 degrees to the lever, at a point 0.4 m from the pivot. Find the moment of the force about the pivot, giving your answer to 3 significant figures.Show answer
Answer: 17.3 N m (50 x 0.4 x sin60)
Q4A uniform beam AB has length 4 m and weight 100 N, resting horizontally on two smooth supports, one at A and one at B. Find the reaction at each support.Show answer
Answer: 50 N at each end (by symmetry, since the weight acts at the midpoint)
Q5A uniform beam AB has length 6 m and weight 150 N. It rests horizontally on two smooth supports, one at C, 1 m from A, and one at D, 2 m from B. Find the reactions at C and D.Show answer
Answer: R_C = 50 N, R_D = 100 N (moments about C: R_D x 3 = 150 x 2)
Q6A uniform rod AB has length 2 m and weight 80 N. The rod is smoothly hinged to a wall at A and held horizontal by a light string attached at B, making an angle of 40 degrees with the rod. Find the tension in the string, giving your answer to 3 significant figures.Show answer
Answer: T = 62.2 N (moments about A: T sin40 x 2 = 80 x 1)
Exam-style questions
Written in the style of a A Level Maths exam paper, with a full mark scheme.
A gate is fitted with a single hinge. A gust of wind exerts a force of 80 N at right angles to the gate, at a point 1.2 m from the hinge. The hinge can safely withstand a maximum moment of 100 N m. (a) Calculate the moment of the 80 N force about the hinge. (b) Determine, with a reason, whether the hinge is at risk of damage.
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A uniform beam AB has length 6 m and weight 210 N. The beam rests horizontally in equilibrium on two smooth supports, one at C, 1 m from A, and one at D, 1.5 m from B. By taking moments about C, find the reaction at D, then find the reaction at C.
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A uniform rod AB has length 2.4 m and weight 96 N. The rod is smoothly hinged to a vertical wall at A and held in a horizontal position by a light inextensible string attached to the rod at B, making an angle of 45 degrees with the rod. A lamp of weight 24 N hangs from a hook at B. (a) By taking moments about A, find the tension in the string, giving your answer to 3 significant figures. (b) Find the magnitude of the vertical component of the force exerted on the rod by the hinge at A.
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