A horizontal board weighing 227.0 N is supported at points A and B in the figure below. The plank serves as a platform for a painter, who weighs 553.0 N. Find the maximum distance D, such that the plank will not tip, no matter where the painter stands. (Let L = 3.80 m.)
There is not "figure beloiw". Try setting the total of moments equal to zero. The weight of the board acts through the center of mass.
To find the maximum distance D, we need to determine the point at which the plank will start tipping.
To begin, let's analyze the forces acting on the plank. We have the weight of the plank itself, which acts downwards at its center of gravity. This force can be represented by a single force, F_plank, located in the middle of the plank.
Next, we have the weight of the painter, which acts downwards at the position D. This force can be represented by a single force, F_painter, located at point D.
Since the plank is not tipping, the sum of the clockwise moments (torques) must be equal to the sum of the counterclockwise moments.
The moment generated by the weight of the plank is given by:
M_plank = F_plank * (L/2)
The moment generated by the weight of the painter is given by:
M_painter = F_painter * (L - D)
Since the plank is not tipping, we can set the moments equal to each other:
M_plank = M_painter
F_plank * (L/2) = F_painter * (L - D)
Now, let's substitute the values into this equation:
227.0 N * (3.80 m/2) = 553.0 N * (3.80 m - D)
To solve for D, let's simplify the equation and isolate D:
(227.0 N * 3.80 m) / 2 = 553.0 N * 3.80 m - 553.0 N * D
427.1 N*m = 2101.4 N*m - 553.0 N * D
553.0 N * D = 2101.4 N*m - 427.1 N*m
553.0 N * D = 1674.3 N*m
D = 1674.3 N*m / 553.0 N
D ≈ 3.02 m
Therefore, the maximum distance D such that the plank will not tip, no matter where the painter stands, is approximately 3.02 m.