Question

Two monkeys off mass 10 kg and 8 kg are moving along a vertical light rope the former climbing up with an acceleration of 2 m/second square while the latter coming down with a uniform velocity of 2 m/sec square find the tension in the rope at the fixed support

Answer 1

Concepts:

Newton's second law, Tension, Free body diagram

Explanation:

To find the tension in the rope at the fixed support, we need to consider the forces acting on each monkey separately and then sum the tensions. For the monkey climbing up, we use Newton's second law to find the tension in the rope. For the monkey coming down with uniform velocity, the tension is equal to its weight since there is no acceleration.

Step by Step Solution:

Step 1

Identify the forces acting on the monkey climbing up (mass = 10 kg, acceleration = 2 m/s^2).

Step 2

Using Newton's second law, the net force on the climbing monkey is given by:
\( F_{net} = T_1 - m_1g = m_1a \)
where \( T_1 \) is the tension in the rope, \( m_1 \) is the mass of the monkey, \( g \) is the acceleration due to gravity (9.8 m/s^2), and \( a \) is the acceleration.

Step 3

Rearrange the equation to solve for \( T_1 \):
\( T_1 = m_1(g + a) = 10 \text{ kg} \times (9.8 \text{ m/s}^2 + 2 \text{ m/s}^2) = 10 \text{ kg} \times 11.8 \text{ m/s}^2 = 118 \text{ N} \)

Step 4

Identify the forces acting on the monkey coming down with uniform velocity (mass = 8 kg). Since the monkey is moving with uniform velocity, there is no acceleration, and the tension \( T_2 \) is equal to the weight of the monkey:
\( T_2 = m_2g = 8 \text{ kg} \times 9.8 \text{ m/s}^2 = 78.4 \text{ N} \)

Step 5

The total tension in the rope at the fixed support is the sum of the tensions due to both monkeys:
\( T_{total} = T_1 + T_2 \)

Step 6

Calculate the total tension:
\( T_{total} = 118 \text{ N} + 78.4 \text{ N} = 196.4 \text{ N} \)

Final Answer:

The tension in the rope at the fixed support is 196.4 N.