Question

A ladder is hanging from ceiling as shown in figure. Three men A, B and C of masses 40 kg, 60 kg, and 50 kg are climbing the ladder. Man A is going up with retardation 2 m/s², C is going up with an acceleration of 1 m/s² and man B is going up with a constant speed of 0.5 m/s. Find the tension in the string supporting the ladder. [g = 9.8 m/s²]

Answer 1

1440 N

Answer 2

To find the tension in the string supporting the ladder, we consider the entire system consisting of the ladder and the three men. We assume the ladder has negligible mass, as its mass is not provided.

The tension in the string must support the effective weight of the men. The effective weight (or apparent weight) of a person of mass 'm' accelerating with 'a' (upwards positive) is given by $F = m(g+a)$. This is the force the person exerts on the support (in this case, the ladder). By Newton's third law, this is also the force the ladder exerts on the person.

Let's calculate the force exerted by each man on the ladder:

1. Man A:

   • Mass ($m_A$) = 40 kg
   • Acceleration ($a_A$) = -2 m/s² (going up with retardation means acceleration is downwards)
   • Force exerted by A on ladder ($F_A$) = $m_A(g + a_A)$
   • $F_A = 40 \text{ kg} \times (9.8 \text{ m/s}^2 - 2 \text{ m/s}^2)$
   • $F_A = 40 \text{ kg} \times 7.8 \text{ m/s}^2$
   • $F_A = 312 \text{ N}$

2. Man B:

   • Mass ($m_B$) = 60 kg
   • Acceleration ($a_B$) = 0 m/s² (going up with constant speed)
   • Force exerted by B on ladder ($F_B$) = $m_B(g + a_B)$
   • $F_B = 60 \text{ kg} \times (9.8 \text{ m/s}^2 + 0 \text{ m/s}^2)$
   • $F_B = 60 \text{ kg} \times 9.8 \text{ m/s}^2$
   • $F_B = 588 \text{ N}$

3. Man C:

   • Mass ($m_C$) = 50 kg
   • Acceleration ($a_C$) = 1 m/s² (going up with acceleration)
   • Force exerted by C on ladder ($F_C$) = $m_C(g + a_C)$
   • $F_C = 50 \text{ kg} \times (9.8 \text{ m/s}^2 + 1 \text{ m/s}^2)$
   • $F_C = 50 \text{ kg} \times 10.8 \text{ m/s}^2$
   • $F_C = 540 \text{ N}$

The total tension (T) in the string supporting the ladder is the sum of the forces exerted by the men on the ladder, assuming the ladder itself has negligible mass and is not accelerating.

$T = F_A + F_B + F_C$ $T = 312 \text{ N} + 588 \text{ N} + 540 \text{ N}$ $T = 1440 \text{ N}$