A uniform 4m plank with a mass of 45 kg is supported at its ends by two ladders. Two 75 kg men are
standing on the plank. One is 1 m from one end and the other is 1.5 m from the other end. Determine the forces exerted by the ladders.
To determine the forces exerted by the ladders, we need to consider the forces acting on the plank.
First, let's consider the forces acting on the plank due to the weight of the men.
The force due to the weight of a person can be calculated using the formula:
force = mass * gravity
where the mass is the mass of the person and gravity is the acceleration due to gravity (9.8 m/s^2).
For the person standing 1 m from one end of the plank:
force1 = mass1 * gravity
= 75 kg * 9.8 m/s^2
= 735 N
For the person standing 1.5 m from the other end of the plank:
force2 = mass2 * gravity
= 75 kg * 9.8 m/s^2
= 735 N
Now, let's consider the forces acting on the plank due to its own weight.
The weight of the plank can be calculated using the formula:
weight = mass * gravity
where the mass is the mass of the plank.
weight = 45 kg * 9.8 m/s^2
= 441 N
Since the plank is uniform and symmetrical, the weight of the plank can be considered to act at its center point.
Next, we can consider the forces acting on the plank due to the ladders.
Since the plank is in equilibrium (not accelerating), the sum of the forces acting on it, both vertically and horizontally, must be zero.
Vertically, the forces acting on the plank can be calculated as:
R1 + R2 - weight - force1 - force2 = 0
where R1 and R2 are the forces exerted by the ladders.
To solve this equation, we need to determine the distances between the ladders and the forces exerted by the ladders.
The distance between the ladders can be calculated by adding the distances of the two men from the ends of the plank:
distance between ladders = 1 m + 1.5 m
= 2.5 m
To determine the forces exerted by the ladders, we can use the principle of moments or torques.
The principle of moments states that for an object in equilibrium, the sum of the clockwise moments about any point is equal to the sum of the anticlockwise moments about the same point.
Using this principle, we can calculate the forces exerted by the ladders as follows:
R1 * (2.5 m) - force1 * (1 m) - weight * (2 m) - force2 * (2.5 m) = 0
Simplifying the equation, we get:
2.5R1 - force1 - 2weight - 2.5force2 = 0
Putting all the values into the equation, we can solve for R1:
2.5R1 - 735 N - 2(441 N) - 2.5(735 N) = 0
2.5R1 - 735 N - 882 N - 1837.5 N = 0
2.5R1 = 3441.5 N
R1 = 1376.6 N
Therefore, the force exerted by the ladder supporting the end of the plank where the person is standing 1 m away is approximately 1376.6 N.
To determine the force exerted by the other ladder, we can use the equation:
R2 = weight + force1 + force2 - R1
R2 = 441 N + 735 N + 735 N - 1376.6 N
R2 = 534.4 N
Therefore, the force exerted by the ladder supporting the other end of the plank where the person is standing 1.5 m away is approximately 534.4 N.