A man of mass 61.6 kg stands on a scaffold supported by a vertical rope at each end. The scaffold has a mass of 15.9 kg and is 2.0 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?

Perform a vertical force balance and a moment balance about the left rope. You will be left with two equations in two unknowns (the two rope tensions), which you can then solve. The weight of the scaffold acts as if applied at the center, 1.0 m from each end. Remember that when writing the moment balance equation.

To find the tension in the right rope, we can first calculate the total torque acting on the scaffold.

1. First, let's find the center of mass of the scaffold. The man stands to the right from the middle of the scaffold at a distance one sixth (1/6) of the length from the center. Since the scaffold is 2.0 m long, the distance of the man from the center is (1/6) × 2.0 m = 1/3 m.

2. The center of mass of an object with uniform mass distribution is located at the midpoint of the object. Since the scaffold has a mass of 15.9 kg and is 2.0 m long, the center of mass is located at the midpoint, which is 2.0 m / 2 = 1.0 m from either end of the scaffold.

3. The total torque acting on the scaffold can be calculated as the product of the force (weight) and the perpendicular distance from the pivot point (rope) to the center of mass.

4. Considering the forces acting on the system, the weight of the scaffold (Wscaffold = mass × gravity) and the weight of the man (Wman = mass × gravity) act downwards, and the tension in both ropes, T, act upwards. The torque due to the weight of the scaffold is Wscaffold × Distance = (15.9 kg × 9.8 m/s^2) × 1.0 m, and the torque due to the weight of the man is Wman × Distance = (61.6 kg × 9.8 m/s^2) × (1/3 m).

5. Since the scaffold is in equilibrium, the sum of the torques due to the forces acting on it must be zero. Therefore, the total torque is:

Total Torque = Torque due to scaffold weight + Torque due to man weight

6. Setting up the equation:

(Wscaffold × Distance) + (Wman × Distance) = 0

((15.9 kg × 9.8 m/s^2) × 1.0 m) + ((61.6 kg × 9.8 m/s^2) × (1/3 m)) = 0

7. Solving for the total torque:

(156.42 N) + (644.848 N) = 0

8. The total torque comes out to be -801.268 N∙m. The negative sign indicates that the torques are in opposite directions, as the scaffold is in equilibrium.

9. Since the scaffold is symmetric, the tension in the left rope will be equal to the tension in the right rope.

10. To find the tension in the right rope, we divide the total torque by the distance between the ropes.

Tension(right) = (total torque) / (distance between the ropes)

Tension(right) = -801.268 N∙m / 4.0 m

11. Finally, calculating the tension in the right rope:

Tension(right) = -200.317 N

The tension in the right rope is approximately -200.317 N (negative sign indicates the direction).