A-Level Mathematics October/November 2024 Q1: Two particles, of masses 1.8 kg and 1.2kg, are connected by a light inextensible string…

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  Let the acceleration of the particles be $a \text{ ms}^{-2}$ and the tension in the string be $T \text{ N}$. Let's assume $g = 10 \text{ ms}^{-2}$.

  The heavier particle (1.8 kg) will accelerate downwards, and the lighter particle (1.2 kg) will accelerate upwards.

  Applying Newton's Second Law ($F=ma$) to the 1.8 kg particle: The downward force is its weight, $1.8g$. The upward force is the tension, $T$. The net force is in the direction of acceleration (downwards). $1.8g - T = 1.8a$ --- (1)

  Applying Newton's Second Law to the 1.2 kg particle: The upward force is the tension, $T$. The downward force is its weight, $1.2g$. The net force is in the direction of acceleration (upwards). $T - 1.2g = 1.2a$ --- (2)

  We now have a system of two simultaneous equations. To find $a$, we can add the two equations to eliminate $T$.

  Adding (1) and (2): $(1.8g - T) + (T - 1.2g) = 1.8a + 1.2a$ $0.6g = 3.0a$ $a = \frac{0.6g}{3.0}$ $a = 0.2g$

  Substituting $g=10$: $a = 0.2 \times 10 = 2$

  So, the magnitude of the acceleration is $2 \text{ ms}^{-2}$.

  To find the tension $T$, substitute the value of $a$ into equation (2): $T - 1.2g = 1.2a$ $T = 1.2a + 1.2g$ $T = 1.2(a+g)$ $T = 1.2(2 + 10)$ $T = 1.2(12)$ $T = 14.4$

  So, the tension in the string is $14.4 \text{ N}$.

  Final Answer: Acceleration = $2 \text{ ms}^{-2}$ Tension = $14.4 \text{ N}$

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  How the marks are awarded

  • M1 — Applying Newton's Second Law to at least one of the particles to form an equation of motion. For example, writing either $1.8g - T = 1.8a$ or $T - 1.2g = 1.2a$, even with sign errors, demonstrates the method.
  • A1 — Obtaining both correct equations of motion: $1.8g - T = 1.8a$ and $T - 1.2g = 1.2a$.
  • DM1 — Solving the simultaneous equations for either $a$ or $T$. In the model answer, this is shown by adding the two equations to find $a$ first.
  • A1 — Obtaining both correct final answers: acceleration $a = 2 \text{ ms}^{-2}$ and tension $T = 14.4 \text{ N}$.

  Common mistakes

  • Incorrectly assigning the direction of forces, leading to sign errors in the equations of motion (e.g., writing $T - 1.8g = 1.8a$ for the heavier particle).
  • Considering the system as a whole but using the difference in masses instead of the sum for the total mass term, e.g., (1.8 - 1.2)g = (1.8 - 1.2)a.
  • Mixing up the masses in the equations, for example writing $1.8g - T = 1.2a$.
  • Making a calculation error when solving the simultaneous equations, or when substituting the value of $a$ back to find $T$.

  Examiner tip: For connected particle problems, always apply $F=ma$ to each particle separately, ensuring the direction of acceleration is consistent for the whole system.

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