A window washer is standing on a scaffold supported by a vertical rope at each end. The scaffold weighs 192 N and is 3.4 m long. What is the tension in each rope when the 708-N worker stands 2.40 m from one end?
Smaller tension: ___N
Larger tension: ___N
you know the sum of tensions is equal to weight, that leads to tension 1=(708+192)-tension2
Now sum moments. Let tension1 be the rope on one end.
Summing moments about end 1
192*3.4/2 + 2.40*708-3.4*tension2=0
solve for tension 2, then use the first equation to solve for tension 1. You have the answers.
To determine the tensions in each rope, we can start by considering the forces acting on the scaffold and worker.
Firstly, we have the weight of the scaffold, which is acting vertically downward. The weight of the scaffold is given as 192 N.
Next, we have the weight of the worker, also acting vertically downward. The weight of the worker is given as 708 N.
Since the scaffold is in equilibrium, the sum of the vertical forces acting on it must be zero. Therefore, the sum of the upward forces must equal the sum of the downward forces.
Let's denote the tension in the rope on the left side as T1, and the tension in the rope on the right side as T2. Considering the forces on the scaffold and worker, we can set up the following equation:
T1 + T2 + 192 N + 708 N = 0
Now, since the worker is standing 2.40 m from one end of the scaffold, the scaffold can be considered as a lever. The distance between the worker and one end is given as 2.40 m.
To solve for the tensions in each rope, we can use the principle of moments or torque. The sum of the clockwise moments must equal the sum of the counterclockwise moments. Considering the moments around the left end of the scaffold, we have:
(708 N)(2.40 m) + (192 N)(3.4 m) - T2(3.4 m) = 0
Simplifying this equation, we get:
1699.2 N·m + 652.8 N·m - 3.4 m·T2 = 0
Combine like terms:
2352 N·m - 3.4 m·T2 = 0
Now, we can solve this equation for T2:
3.4 m·T2 = 2352 N·m
T2 = 2352 N·m / 3.4 m
T2 = 692.94 N
Therefore, the tension in the rope on the right side (larger tension) is approximately 692.94 N.
To find the tension in the rope on the left side (smaller tension), we can substitute this value back into the equation we set up earlier:
T1 + 692.94 N + 192 N + 708 N = 0
Combine like terms:
T1 + 1592.94 N = 0
Subtracting 1592.94 N from both sides:
T1 = -1592.94 N
Since tension is a pulling force, the negative sign indicates that T1 is in the opposite direction. To obtain the positive value, we ignore the negative sign. Therefore, the tension in the rope on the left side (smaller tension) is approximately 1592.94 N.
So, the answers are:
Smaller tension: 1592.94 N
Larger tension: 692.94 N