A scaffold (a uniform horizontal board of mass 50 kg and length 4 m) is supported by a rope at each end. A 100 kg painter stands on the scaffold at a location such that the tension is one rope is four times that in the other. Where is the painter standing (specify clearly the distance from one or the other end)?

4 F up at one end, F up at the other

so
100 + 50 = 5 F
F = 30
4 F = 120
take moments around end close to the 100 kg painter. x is from that end (the 4F=120 end) to the painter)

100 x + 2(50) = 4(30)

100 x = 20
x = .2 meters

To find out where the painter is standing on the scaffold, we need to consider the forces acting on the scaffold and set up an equation to solve for the position.

Let's denote the distance of the painter from one end of the scaffold as 'x'. Since the scaffold is uniform, the center of mass will be at the midpoint, which is 2 meters from each end. Therefore, the painter's distance from one end will be (2 + x) meters.

Now, let's consider the forces acting on the scaffold. There are three forces involved: the weight of the scaffold, the weight of the painter, and the tension in each rope.

1. Weight of the scaffold: The weight acts vertically downward from the center of mass, which is at the midpoint of the scaffold's length. The weight can be calculated as follows:
Weight_scaffold = mass_scaffold * g
Weight_scaffold = 50 kg * 9.8 m/s^2 = 490 N

2. Weight of the painter: The weight of the painter acts downward from the location where the painter is standing, which is (2 + x) meters from one end. The weight can be calculated as:
Weight_painter = mass_painter * g
Weight_painter = 100 kg * 9.8 m/s^2 = 980 N

3. Tension in each rope: Let's assume that the tension in one rope is T1 and in the other rope is T2. It is given that the tension in one rope is four times that in the other, so we can write:
T1 = 4 * T2

Now, setting up the equation:

Sum of vertical forces = Weight_scaffold + Weight_painter
T1 + T2 = Weight_scaffold + Weight_painter
4 * T2 + T2 = 490 N + 980 N
5 * T2 = 1470 N
T2 = 294 N

Therefore, the tension in one rope (T1) is 4 * T2 = 4 * 294 N = 1176 N.

Now, we can solve for the position 'x' by considering the torques acting on the scaffold about a chosen axis (let's say the left end of the scaffold).

The torque due to the weight of the scaffold is zero since it acts at the midpoint, which is the axis of rotation.

The torque due to the weight of the painter can be calculated as follows:
Torque_painter = Weight_painter * (2 + x) = 980 N * (2 + x)

The torque due to the tension in one rope can be calculated as:
Torque_T1 = T1 * x = 1176 N * x

Setting up the torque equation:

Torque_painter = Torque_T1
980 N * (2 + x) = 1176 N * x

Simplifying the equation:

1960 N + 980 N * x = 1176 N * x
1960 N = 1176 N * x - 980 N * x
1960 N = 196 N * x
x = 1960 N / 196 N
x = 10 meters

Therefore, the painter is standing 10 meters from one end of the scaffold.