A man of mass 75.9 kg stands on a scaffold supported by a vertical rope at each end. The scaffold has a mass of 19.8 kg and is 2.8 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?

It does not matter how long the plank is, only that he is 1/6 of the length from the middle.

Therefore to make the numbers easy I am going to say the plank is 6 meters long (not 2.8)
He stands one meter right of center. That is 4 meters from the left and 2 meters from the right.
Now take moments about the left end of the plank and call the tension in the left line T.
T (6) = m g (4)
T = (2/3) m g = .667*75.9*9.81 Newtons

To find the tension in the right rope, we need to consider the forces acting on the scaffold.

First, let's calculate the distance from the right end of the scaffold to the point where the man is standing:
Distance from right end = (1/6) * 2.8 m = 0.4667 m

Now, let's consider the forces acting on the scaffold:
1. Weight of the man (acting downward) = mass of the man * acceleration due to gravity = 75.9 kg * 9.8 m/s^2
2. Weight of the scaffold (acting downward) = mass of the scaffold * acceleration due to gravity = 19.8 kg * 9.8 m/s^2
3. Tension in the left rope (acting upward)
4. Tension in the right rope (acting upward)

Since the scaffold is in equilibrium (not accelerating), the sum of the forces in the vertical direction must be zero.
This can be expressed as:
Tension in the left rope + Tension in the right rope - Weight of the man - Weight of the scaffold = 0

Now, let's solve for the tension in the right rope:
Tension in the right rope = Weight of the man + Weight of the scaffold - Tension in the left rope

Substituting the known values:
Tension in the right rope = (75.9 kg * 9.8 m/s^2) + (19.8 kg * 9.8 m/s^2) - Tension in the left rope

However, we still need to find the tension in the left rope. To do this, we need to consider the torques acting on the scaffold.

The torque (τ) acting on an object is given by the equation:
τ = force * distance * sin(θ)

In this case, the torques acting on the scaffold are:

1. Torque due to the weight of the man:
τ_man = (Weight of the man) * (Distance from the point of rotation) * sin(θ)
θ is the angle between the force vector and the distance vector, which is 90 degrees in this case.

2. Torque due to the weight of the scaffold:
τ_scaffold = (Weight of the scaffold) * (Distance from the point of rotation) * sin(θ)
θ is the angle between the force vector and the distance vector, which is 90 degrees in this case.

Since the scaffold is at rest, the net torque acting on it must be zero, which means the sum of the torques must be zero.
This can be expressed as:

τ_man + τ_scaffold = 0

Now, let's calculate the torques:
τ_man = (Weight of the man) * (Distance from the right end of the scaffold to the point where the man is standing)
τ_scaffold = (Weight of the scaffold) * (Distance from the right end of the scaffold to the midpoint)

Setting the torques equal to each other and rearranging the equation, we can solve for the tension in the left rope:
Tension in the left rope = Weight of the man * Distance from the point where the man is standing to the right end of the scaffold / Distance from the midpoint to the right end of the scaffold

Substituting the known values, we can calculate the tension in the left rope.

Finally, substituting the tension in the left rope into the equation for the tension in the right rope, we can calculate the tension in the right rope.