A man of mass 67.1 kg stands on a scaffold supported by a vertical rope at each end. The scaffold has a mass of 24.6 kg and is 3.6 m long. Suppose the man stands to the right from the middle of the scaffold that is a distance one sixth of the length of the scaffold. What is the tension in the right rope?

To determine the tension in the right rope, we need to analyze the forces acting on the scaffold.

1. First, let's find the center of mass of the system. Since the man is standing one-sixth of the length of the scaffold from the right end, the distance from the left end of the scaffold to the center of mass is (5/6) * 3.6 m = 3.0 m.

2. Now, let's consider the forces acting on the system. We have the weight of the man acting downward, which can be calculated as 67.1 kg * 9.8 m/s^2 = 657.98 N. This force acts vertically downward from the center of mass.

3. The weight of the scaffold will act downwards from its center of mass, which is at the midpoint. The weight of the scaffold can be calculated as 24.6 kg * 9.8 m/s^2 = 240.68 N.

4. Since the system is in equilibrium, the total force acting upward must balance the total force acting downward. The tension in the right rope will provide an upward force, opposing the combined weight of the man and the scaffold.

5. The total weight of the system is the sum of the weight of the man and the weight of the scaffold, which is 657.98 N + 240.68 N = 898.66 N.

6. Since the scaffold is in equilibrium, the total force acting upward at the midpoint must be equal to half of the total weight of the system. Therefore, the tension in each rope will be half of the total weight.

7. The tension in the right rope is given by 1/2 * 898.66 N = 449.33 N.

Therefore, the tension in the right rope is 449.33 N.

To find the tension in the right rope, we need to calculate the forces acting on the scaffold and the man and then apply Newton's second law (F = ma) in the vertical direction.

1. Calculate the center of mass of the scaffold:
The man stands to the right from the middle of the scaffold, a distance one-sixth of the length. One-sixth of the length of the scaffold is (1/6) * 3.6 m = 0.6 m. Therefore, the center of mass of the scaffold is at (3.6 / 2) + 0.6 = 1.8 + 0.6 = 2.4 m from the left end of the scaffold.

2. Calculate the gravitational force acting on the scaffold:
The gravitational force acting on an object is given by the formula F = mg, where F is the force in Newtons, m is the mass in kilograms, and g is the acceleration due to gravity (approximately 9.8 m/s^2). The mass of the scaffold is 24.6 kg, so the gravitational force acting on the scaffold is F_scaffold = 24.6 kg * 9.8 m/s^2 = 240.28 N.

3. Calculate the gravitational force acting on the man:
The mass of the man is 67.1 kg. Therefore, the gravitational force acting on the man is F_man = 67.1 kg * 9.8 m/s^2 = 657.98 N.

4. Calculate the torque about the right end of the scaffold:
Torque is the product of force and distance from a point, so we need to calculate the torque about the right end of the scaffold. The torque due to the gravitational force of the scaffold is 240.28 N * 2.4 m = 576.672 Nm. The torque due to the gravitational force of the man is 657.98 N * 3.6 m = 2363.328 Nm. Therefore, the total torque about the right end of the scaffold is 576.672 Nm + 2363.328 Nm = 2940 Nm.

5. Calculate the tension in the right rope:
In equilibrium, the sum of torques about any point is zero. Since the man is standing to the right of the middle of the scaffold, we can consider the torques about the right end of the scaffold. The only torque acting in the clockwi