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
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
1. Identify the forces acting on the rod: the weight (W) of the rod and the vertical force (F).
- S2
2. The weight acts at the center of mass, which is 2 m from either end.
- S3
3. Calculate the moment due to the weight about the hinge: M weight = W × d = 20 N × 2 m = 40 Nm (clockwise).
- S4
4. For equilibrium, the total clockwise moments must equal the total counterclockwise moments.
- S5
5. The moment due to the vertical force about the hinge is M force = F × d = F × 4 m (counterclockwise).
- S6
6. Set up the equation for equilibrium: M weight = M force → 40 Nm = F × 4 m.
- S7
7. Solve for F: F = 40 Nm ÷ 4 m = 10 N.
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
1. Identify the forces acting on the rod: the weight (W) of the rod and the vertical force (F).
- S2
2. The weight acts at the center of mass, which is 2 m from either end.
- S3
3. Calculate the moment due to the weight about the hinge: M weight = W × d = 20 N × 2 m = 40 Nm (clockwise).
- S4
4. For equilibrium, the total clockwise moments must equal the total counterclockwise moments.
- S5
5. The moment due to the vertical force about the hinge is M force = F × d = F × 4 m (counterclockwise).
- S6
6. Set up the equation for equilibrium: M weight = M force → 40 Nm = F × 4 m.
- S7
7. Solve for F: F = 40 Nm ÷ 4 m = 10 N.
Final answer: The magnitude of the vertical force required to keep the rod horizontal is 10 N.
Common mistakes
- Forgetting to consider the direction (clockwise or counterclockwise) of moments. — 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. — 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. — List all forces acting on the system and their points of application before calculating moments.
- Incorrectly setting up the equilibrium equation. — 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. — Always verify that both conditions for static equilibrium are satisfied: ΣF = 0 and ΣM = 0.
- Using incorrect units for moments. — Always use the correct units for moments: Newton-metres (Nm).
- Misinterpreting the problem statement. — Read the problem carefully and identify all given information before starting the solution.
- Not simplifying the final answer. — Always simplify your final answer and express it in the required units.
Where the marks go
- Full worked solution (all marking points)6 marks