Concepts:
Newton's second law, Tension, Free body diagram
Explanation:
To determine the rope tension at the fixed support, we must analyze the forces on each monkey individually and then sum the tensions. For the ascending monkey, Newton's second law is applied to calculate the rope tension. For the descending monkey moving at a constant velocity, the tension equals its weight due to the absence of acceleration.
Step by Step Solution:
Step 1
Identify forces on the climbing monkey (mass = 10 kg, acceleration = 2 m/s^2).
Step 2
Apply Newton's second law for the net force on the climbing monkey:
\( F_{net} = T_1 - m_1g = m_1a \)
where \( T_1 \) is rope tension, \( m_1 \) is monkey mass, \( g \) is gravitational acceleration (9.8 m/s^2), and \( a \) is acceleration.
Step 3
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 forces on the descending monkey with uniform velocity (mass = 8 kg). Since velocity is uniform, acceleration is zero, and tension \( T_2 \) equals the monkey's weight:
\( T_2 = m_2g = 8 \text{ kg} \times 9.8 \text{ m/s}^2 = 78.4 \text{ N} \)
Step 5
The total rope tension at the fixed support is the sum of tensions from 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.