For M.A. by Tutoring the nation A - Level Mechanics Connected Particles over a Pulley A complete step - by - step tutorial May 2018 A - Level Exam · Question 9 For M.A. by Tutoring the nation This is called an Atwood machine — one of the most common set - ups in A - Level Mechanics. Two balls, P and Q, hang from the ends of a single string that passes over a pulley fixed to the ceiling. Ball P has mass 2m and ball Q has mass km, where k < 2, meanin g P is heavier. When the system is released, P falls and Q rises. The problem tells us P accelerates downward at 5g/7. We use this to find the tension in the string (part a), explain Q’s acceleration (part b), and find k (part c). Key Terminology — read this first! Mass (m): How much matter is in an object. Measured in kg. We leave it as 'm' because it cancels later. Weight (mg): The downward gravitational force. Weight = mass × g, where g ≈ 9.8 m/s². Tension (T): The upward pulling force in a string. It acts upward on any hanging ball. Acceleration (a): How quickly speed changes. Here a = 5g/7, meaning the balls speed up at five - sevenths of g. Newton’s Second Law: Net Force = mass × acceleration, written as F = ma. Inextensible string: Cannot stretch — so both balls always move at the same speed and magnitude of acceleration. Smooth pulley: No friction — tension is the same on both sides of the string. Particle model: Treat balls as point masses with no size, ignoring effects like spinning or air resistance. The Question (a) Find, in terms of m and g, the tension in the string. [3 marks] (b) Explain why the acceleration of Q also has magnitude 5g/7. [1 mark] (c) Find the value of k. [4 marks] (d) Identify one limitation of the model that will affect the accuracy of your answer to part (c). [1 mark] What Is This Problem About? For M.A. by Tutoring the nation Strategy We focus on ball P alone. We know: mass = 2m, acceleration = 5g/7 downward, weight = 2mg downward, tension = T upward. Apply Newton’s Second Law, taking downward as positive (the direction P moves). Step - by - step working: 1 Identify forces on P: • Weight downward: 2mg • Tension upward: T 2 Take downward as positive (direction P moves). 3 Write Newton’s Second Law: (Net force) = mass × acceleration 2mg − T = 2m × (5g/7) 4 Expand the right - hand side: 2mg − T = 10mg/7 5 Rearrange: T = 2mg − 10mg/7 Write 2mg as 14mg/7 so we can subtract: T = 14mg/7 − 10mg/7 6 Subtract the fractions: T = 4mg/7 T = 4mg / 7 Why ‘in terms of m and g’? The question asks for T using the letters m and g, not a decimal. This is because m was never given a specific value. Our answer T = 4mg/7 is true for any mass m — it works universally. Part (b) — Why Does Q Also Accelerate at 5g/7? [1 mark] What the examiner wants This is a one - mark explain question. One clear sentence using the key physics word earns the mark. No calculation needed. Model answer: Because the string is inextensible (it cannot stretch), both ends of the string must always move at the same speed. Therefore both balls must always have the same magnitude of acceleration. Key word: inextensible — always use this word in your answer. Part (a) — Find the Tension T [3 marks] For M.A. by Tutoring the nation Part (c) — Find the Value of k [4 marks] Strategy Now apply Newton’s Second Law to ball Q. We know: mass = km, acceleration = 5g/7 upward, tension = T = 4mg/7 (from part a). Taking upward as positive (direction Q moves). Step - by - step working: 1 Identify forces on Q: • Tension upward: T • Weight downward: kmg 2 Take upward as positive (direction Q moves). 3 Write Newton’s Second Law for Q: T − kmg = km × (5g/7) 4 Substitute T = 4mg/7: 4mg/7 − kmg = 5kmg/7 5 Divide every term by mg (m ≠ 0, g ≠ 0, so this is valid): 4/7 − k = 5k/7 6 Multiply every term by 7 to remove fractions: 4 − 7k = 5k 7 Add 7k to both sides: 4 = 12k 8 Divide both sides by 12: k = 4/12 = 1/3 k = 1/3 (≈ 0.333...) Check: does k < 2? The question states k < 2. Since 1/3 ≈ 0.33 < 2, our answer is consistent. Always verify your answer against given conditions. For M.A. by Tutoring the nation A limitation is an assumption in the model that is unrealistic, causing inaccuracy. You only need one for the mark. Good answers (any one earns the mark): Assumption Why it is unrealistic Effect on k Pulley is smooth (no friction) Real pulleys have friction, so tension differs on each side of the string. k would change String is light (massless) A real string has mass which contributes weight and affects motion. k would change Balls are particles (no size) Real balls have size; air resistance and shape effects act on them. k would change Part (d) — Limitation of the Model [1 mark] For M.A. by Tutoring the nation Part Result Marks (a) Tension T = 4mg/7 3 (b) Acceleration of Q Same acceleration — the string is inextensible 1 (c) Value of k k = 1/3 4 (d) Limitation e.g. Pulley not smooth / string has mass / balls have size 1 Total 9 Glossary of Key Terms Acceleration How quickly velocity changes. Measured in m/s². Atwood Machine Two masses on a string over a pulley. Used to study forces and acceleration. Force A push or pull. Measured in Newtons (N). Forces change an object’s motion. Gravitational field strength (g) Force of gravity per kilogram ≈ 9.8 N/kg near Earth’s surface. Inextensible Cannot stretch. Both ends of the string move at the same speed and acceleration. Light string Modelled as having zero mass so its weight doesn’t affect the motion. Net / Resultant force The total force after all individual forces are combined with direction. Newton’s Second Law F = ma: net force = mass × acceleration. Applied separately to each object. Particle A point mass with no size. Eliminates complications like rotation and drag. Smooth pulley No friction, so tension is the same on both sides of the string. Tension The pulling force in a string. Always acts upward on a hanging ball. Weight Gravitational force = mass × g. Always acts downward. Summary of Results For M.A. by Tutoring the nation Exam Tips for Connected Particle Questions 1. Draw a clear diagram first. Label every force on each object separately. Don’t try to do it in your head. 2. Apply F = ma to each object separately. Write one equation for P and one for Q. Never combine unless you are very confident. 3. State your positive direction. Say which direction you take as positive for each object. This avoids sign errors. 4. Use answers from earlier parts. In part (c), you must substitute T = 4mg/7. Forgetting is a very common mistake. 5. Simplify before solving. Divide through by m or g early — it makes the algebra much cleaner. 6. Always check against given conditions. If the question states k < 2, verify your answer satisfies this. A - Level Mechanics Tutorial · Connected Particles · May 2018 · Question 9