A house painter is standing on a uniform, horizontal platform that is held in equilibrium by two cables attached to supports on the roof. The painter has a mass of 75 kg and the mass of the platform is 20.0 kg. The distance from the left end of the platform to where the painter is standing is d = 1.9 m and the total length of the platform is 4.9 m.
To solve this problem, we need to analyze the forces acting on the painter and the platform.
1. Gravity force: Both the painter and the platform experience the force of gravity pulling them downwards. The force of gravity is given by the equation F_gravity = m * g, where m is the mass and g is the acceleration due to gravity (9.8 m/s^2).
For the painter:
F_gravity_painter = 75 kg * 9.8 m/s^2 = 735 N
For the platform:
F_gravity_platform = 20 kg * 9.8 m/s^2 = 196 N
2. Tension forces: The tension forces in the cables counterbalance the force of gravity, keeping the platform in equilibrium. Let's call the tension in the left cable T_left and the tension in the right cable T_right.
3. Horizontal forces: Since the platform is in equilibrium, the sum of the horizontal forces is zero. This means that the tension forces in the cables cancel each other out, resulting in no net horizontal force. Therefore, T_left = T_right.
Vertical forces:
By analyzing the vertical forces, we can find the value of the tension forces in the cables.
Let's assume the distance between the left side of the platform and the point where the left cable attaches to the roof is x.
The torque exerted by gravity on the painter around the point where the left cable attaches to the roof is given by the equation Torque_left = F_gravity_painter * (d - x). This torque creates a clockwise rotation.
The torque exerted by gravity on the platform is given by the equation Torque_platform = F_gravity_platform * (d - (x + 4.9 - d)). This torque creates a counterclockwise rotation.
Since the platform is in equilibrium, the torques created by these forces should be equal, so we can set up the following equation:
Torque_left = Torque_platform
F_gravity_painter * (d - x) = F_gravity_platform * (d - (x + 4.9 - d))
Simplifying the equation, we have:
735 N * (d - x) = 196 N * (d - (x + 4.9 - d))
Solving this equation will give us the value of x, which represents the distance from the left side of the platform to the point where the left cable attaches to the roof.
Once we have x, we can calculate the tension in the cables using the equation for horizontal equilibrium:
T_left = T_right = F_gravity_painter + F_gravity_platform
Substituting the values, we get:
T_left = T_right = 735 N + 196 N = 931 N
So, the tension in each cable is 931 N.