Connected Bodies - The Lift Problem (Edexcel A-Level Mathematics): Revision Notes

• Scenario: Consider a lift of mass $M$ moving vertically with an acceleration $a$. Inside the lift is a person or object of mass $m$.
• Objective: Determine the tension in the cable supporting the lift and the apparent weight of the person or object inside the lift under various conditions.

1. On the Lift:

• Weight of the Lift: $Mg$ (acting downward).
• Tension in the Cable: $T$ (acting upward).
• Net Force on the Lift: Determines the acceleration of the lift.

2. On the Person/Object Inside the Lift:

• Weight of the Person/Object: $mg$ (acting downward).
• Normal Force (Apparent Weight): $N$ (acting upward). This is the force exerted by the floor of the lift on the person or object.

Using Newton's second law, $F = ma$ , for the lift:

$T - Mg = Ma$

$T = M(g + a)$

Where:

• $T$ is the tension in the cable.
• $M$ is the mass of the lift.
• $a$ is the acceleration of the lift.

Example 1: Lift Accelerating Upwards

• Given:
• Mass of lift $M = 500 \ , \text{kg}$
• Mass of person $m = 70 \ , \text{kg}$
• Acceleration $a = 2 \ , \text{m/s}^2$ (upwards)
• Find:
• Tension in the cable $T$
• Apparent weight of the person $N$
• Solution:

1. Tension in the Cable:

$T = M(g + a) = 500 \times (9.8 + 2) = 500 \times 11.8 = :success[5900 \, \text{N}]$

2. Apparent Weight of the Person:

$N = m(g + a) = 70 \times (9.8 + 2) = 70 \times 11.8 = :success[826 \, \text{N}]$

Example 2: Lift Accelerating Downwards

• Given:
• Mass of lift $M = 500 \, \text{kg}$
• Mass of person $m = 70 \, \text{kg}$
• Acceleration $a = 3 \, \text{m/s}^2$ (downwards)
• Find:
• Tension in the cable $T$
• Apparent weight of the person $N$
• Solution:

3. Tension in the Cable:

$T = M(g - a) = 500 \times (9.8 - 3) = 500 \times 6.8 = :success[3400 \, \text{N}]$

4. Apparent Weight of the Person:

$N = m(g - a) = 70 \times (9.8 - 3) = 70 \times 6.8 = :success[476 \, \text{N}]$

The lift problem involves applying Newton's second law to both the lift and the person or object inside it. By analysing the forces acting on the system, you can determine the tension in the cable and the apparent weight of the person or object under various conditions of acceleration. Understanding these principles is crucial for solving more complex problems involving connected bodies in A-Level Mechanics.