A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Newton’s Third Law and Equilibrium of Forces — mark scheme explained
The short answer
In this section, we will explore Newton's third law, the equilibrium of forces on a particle, motion in a straight line under specific conditions, problems involving smooth pulleys and connected particles, resolving forces in two dimensions, and the equilibrium of a particle under coplanar forces.
The question
A particle of mass 2 kg is acted upon by two forces: F 1 = 5 N at 30° to the horizontal and F 2 = 8 N horizontally. Determine if the particle is in equilibrium.
[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 equilibrium problems, marks are often awarded for correctly resolving forces, setting up equations, and solving them. Ensure you show all steps clearly and check your units. For connected particle problems, marks are typically given for applying Newton's laws to each mass, solving simultaneous equations, and providing the final answer.
What the command words demand
- Determine
- Calculate or find a specific value or condition.
- Explain
- Provide a clear and detailed reasoning for a concept or phenomenon.
- Resolve
- Break down a force into its components along specified axes.
- Verify
- Check the correctness of a statement or solution using given data.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 5-7 minutes per question in this section to ensure you have enough time to resolve forces, set up equations, and solve them accurately.
- Resolve F 1 into its horizontal and vertical components: F 1x = 5 cos 30° = 4.33 N, F 1y = 5 sin 30° = 2.5 N.1 mark
- The horizontal component of the total force is ΣF x = F 1x + F 2 = 4.33 + 8 = 12.33 N.1 mark
- The vertical component of the total force is ΣF y = F 1y = 2.5 N.1 mark
- Since ΣF x ≠ 0 and ΣF y ≠ 0, the particle is not in equilibrium.1 mark
Final answer: The particle is not in equilibrium.
Work through every step correctly and you earn all 4 marks.
Another worked example
Two masses m 1 = 3 kg and m 2 = 5 kg are connected by a light, inextensible string over a smooth pulley. Determine the acceleration of the system and the tension in the string.
- Apply Newton's second law to each mass: T - m 1 g = m 1 a and m 2 g - T = m 2 a.2 marks
- Substitute the values: T - 3 × 9.8 = 3a and 5 × 9.8 - T = 5a.1 mark
- Simplify the equations: T - 29.4 = 3a and 49 - T = 5a.0 marks
- Add the two equations to eliminate T: (T - 29.4) + (49 - T) = 3a + 5a → 19.6 = 8a → a = 2.45 m/s 2 .2 marks
- Substitute a back into one of the original equations to find T: T - 29.4 = 3 × 2.45 → T = 36.75 N.1 mark
Final answer: The acceleration is 2.45 m/s 2 , and the tension in the string is 36.75 N.
Work through every step correctly and you earn all 6 marks.
Common mistakes
Forgetting to resolve forces into components
Why it happens: Students often overlook the need to break down forces into their horizontal and vertical components when dealing with problems in two dimensions.
Fix: Always check if the problem involves forces at angles. If so, resolve each force into its x and y components before applying equilibrium conditions or Newton's laws.
Incorrectly applying Newton's third law
Why it happens: Students sometimes confuse the direction of action-reaction pairs, leading to incorrect force diagrams.
Fix: Remember that for every action, there is an equal and opposite reaction. Ensure that the forces are drawn in the correct directions on your diagram.
Forgetting to consider all forces acting on a particle
Why it happens: Students may overlook certain forces, such as friction or tension, when setting up their equations.
Fix: Always list all the forces acting on the particle and ensure they are included in your calculations. Double-check your force diagram.
Incorrectly calculating the resultant force
Why it happens: Students may add or subtract forces incorrectly, especially when dealing with vector quantities.
Fix: Use vector addition to find the resultant force. Ensure you are adding and subtracting components correctly.
Confusing equilibrium conditions
Why it happens: Students may mix up the conditions for equilibrium in different dimensions, leading to incorrect equations.
Fix: Remember that a particle is in equilibrium if ΣF x = 0 and ΣF y = 0. Ensure you are applying these conditions correctly.
Incorrectly solving simultaneous equations
Why it happens: Students may make algebraic errors when solving for tension or acceleration in connected particle problems.
Fix: Double-check your algebra and ensure you are solving the equations correctly. It can be helpful to substitute values back into the original equations to verify your solution.
Forgetting to check units
Why it happens: Students may forget to convert units or ensure consistency in their calculations, leading to incorrect answers.
Fix: Always check that all forces are in the same unit (e.g., Newtons) and that masses are in kilograms. Ensure your final answer has the correct units.
Incorrectly interpreting force diagrams
Why it happens: Students may misinterpret the direction or magnitude of forces on a diagram, leading to incorrect equations.
Fix: Take your time to draw and label the force diagram accurately. Ensure you understand the direction and magnitude of each force before setting up your equations.
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 |
|---|---|---|
| Equilibrium Of Forces | Resolve two forces into components and test whether the particle is in equilibrium. | 4 |
| Connected Particles | Apply Newton's second law to two masses on a pulley to find acceleration and tension. | 6 |
| Force Equilibrium Test | Resolve the three forces into components to determine whether the particle is in equilibrium. | 4 |
| Equilibrium Of Forces | Resolve the two forces into components and check whether the resultant is zero. | 4 |
| Total across these question types | 18 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.