A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Moments in Simple Static Contexts — mark scheme explained
The short answer
Moments are a fundamental concept in mechanics, particularly when dealing with static equilibrium. A moment is the turning effect of a force around a point or an axis. Understanding and using moments in simple static contexts is crucial for solving problems involving levers, beams, and other mechanical systems.
The question
A uniform rod of length 4 m and weight 20 N is supported at one end by a hinge and at the other end by a vertical force. Calculate the magnitude of the vertical force required to keep the rod horizontal.
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For questions involving moments, marks are typically awarded for correctly identifying forces and distances, setting up the equilibrium equations, performing calculations, and providing the final answer. Partial credit may be given for correct steps even if the final answer is incorrect.
What the command words demand
- Calculate
- Perform a numerical calculation to find a specific value.
- Determine
- Find or establish a particular value or quantity.
- Explain
- Provide a clear and detailed account of how something works or why it happens.
- Identify
- Recognize and name key elements, forces, or principles in a problem.
- Show
- Demonstrate a step-by-step process to arrive at a given result.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate approximately 5-7 minutes per question to ensure you have enough time to carefully identify all forces, calculate moments, and check your work.
- 1. Identify the forces acting on the rod: the weight (W) of the rod and the vertical force (F).1 mark
- 2. The weight acts at the center of mass, which is 2 m from either end.1 mark
- 3. Calculate the moment due to the weight about the hinge: M weight = W × d = 20 N × 2 m = 40 Nm (clockwise).1 mark
- 4. For equilibrium, the total clockwise moments must equal the total counterclockwise moments.0 marks
- 5. The moment due to the vertical force about the hinge is M force = F × d = F × 4 m (counterclockwise).1 mark
- 6. Set up the equation for equilibrium: M weight = M force → 40 Nm = F × 4 m.1 mark
- 7. Solve for F: F = 40 Nm ÷ 4 m = 10 N.1 mark
Final answer: The magnitude of the vertical force required to keep the rod horizontal is 10 N.
Work through every step correctly and you earn all 6 marks.
Another worked example
A uniform beam of length 6 m and weight 30 N is supported by two vertical forces, one at each end. Calculate the magnitudes of these forces if the beam is in equilibrium.
- 1. Identify the forces acting on the beam: the weight (W) of the beam and the two vertical forces (F 1 and F 2 ).0 marks
- 2. The weight acts at the center of mass, which is 3 m from either end.1 mark
- 3. For equilibrium, the sum of all forces must be zero: F 1 + F 2 - W = 0 → F 1 + F 2 = 30 N.1 mark
- 4. Calculate the moment due to the weight about one end (say, the left end): M weight = W × d = 30 N × 3 m = 90 Nm (clockwise).1 mark
- 5. The moment due to F 2 about the left end is M F2 = F 2 × 6 m (counterclockwise).1 mark
- 6. For equilibrium, the total clockwise moments must equal the total counterclockwise moments: M weight = M F2 → 90 Nm = F 2 × 6 m.1 mark
- 7. Solve for F 2 : F 2 = 90 Nm ÷ 6 m = 15 N.2 marks
- 8. Substitute F 2 into the force equilibrium equation: F 1 + 15 N = 30 N → F 1 = 15 N.1 mark
Final answer: The magnitudes of the vertical forces are F 1 = 15 N and F 2 = 15 N.
Work through every step correctly and you earn all 8 marks.
Common mistakes
Forgetting to consider the direction (clockwise or counterclockwise) of moments.
Why it happens: Students often forget that moments have a direction, and this can lead to incorrect calculations when summing moments for equilibrium.
Fix: Always label the direction of each moment and use the correct sign (positive for counterclockwise, negative for clockwise).
Using the wrong distance in the moment calculation.
Why it happens: Students sometimes use the total length of the beam or rod instead of the perpendicular distance from the pivot to the line of action of the force.
Fix: Always identify and measure the perpendicular distance from the pivot to the point where the force acts.
Not considering all forces acting on the system.
Why it happens: Students may overlook some forces, such as the weight of the beam or rod, which can lead to incorrect moment calculations.
Fix: List all forces acting on the system and their points of application before calculating moments.
Incorrectly setting up the equilibrium equation.
Why it happens: Students may set up the equilibrium equation incorrectly, leading to incorrect solutions.
Fix: Ensure that the sum of all clockwise moments equals the sum of all counterclockwise moments (ΣM = 0).
Forgetting to check if the system is in static equilibrium.
Why it happens: Students may solve for one condition (e.g., moment equilibrium) but forget to check the other condition (force equilibrium).
Fix: Always verify that both conditions for static equilibrium are satisfied: ΣF = 0 and ΣM = 0.
Using incorrect units for moments.
Why it happens: Students may use the wrong units, such as Newtons (N) instead of Newton-metres (Nm).
Fix: Always use the correct units for moments: Newton-metres (Nm).
Misinterpreting the problem statement.
Why it happens: Students may misread or misunderstand the problem, leading to incorrect setup and calculations.
Fix: Read the problem carefully and identify all given information before starting the solution.
Not simplifying the final answer.
Why it happens: Students may leave their answers in a complex form, which can lead to marks being lost for not providing a simplified final answer.
Fix: Always simplify your final answer and express it in the required units.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Principle Of Moments | Apply the principle of moments to find the vertical force keeping a hinged rod horizontal. | 6 |
| Moments And Equilibrium | Find two support forces on a beam using force and moment balance. | 8 |
| Moments And Equilibrium | Take moments about the hinge to find the vertical force keeping the rod horizontal. | 7 |
| Principle Of Moments | Take moments about the hinge to find the vertical support force keeping the beam horizontal. | 8 |
| Total across these question types | 29 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.