a 200lb pound man walks out to the end of a uniform plank which projects perpendiculary over the edge of a roof. the plank is 20 ft long and weights 100lb. how far from the edge of the roof can the plank overhang?
To determine how far the plank can overhang, we need to consider the balance of torques acting on it.
First, let's calculate the torque exerted by the man (M) at the end of the plank. The weight of the man can be considered acting at the center of gravity, which is halfway between the man's feet and the end of the plank.
Torque exerted by the man, T_man = weight of the man x distance between the center of gravity and the end of the plank.
Given that the man weighs 200 lb and the plank is 20 ft long:
T_man = 200 lb x (20/2) ft
T_man = 200 lb x 10 ft
T_man = 2000 lb*ft
Next, let's calculate the torque exerted by the plank itself. The weight of the plank can be considered acting at its center of gravity, which is at the midpoint.
Torque exerted by the plank, T_plank = weight of the plank x distance between the center of gravity and the end of the plank.
Given that the plank weighs 100 lb and is 20 ft long:
T_plank = 100 lb x (20/2) ft
T_plank = 100 lb x 10 ft
T_plank = 1000 lb*ft
In order for the system to be in equilibrium, the total torque exerted by the man must be equal and opposite to the torque exerted by the plank:
T_man = -T_plank
2000 lb*ft = -1000 lb*ft
As the torques balance out, we can deduce that the plank can overhang the edge of the roof by 20 feet.
To determine how far from the edge of the roof the plank can overhang, we need to consider the equilibrium of forces acting on the system.
Let's break down the forces involved:
1. Weight of the man: The man has a weight of 200 lb, acting vertically downward.
2. Weight of the plank: The plank weighs 100 lb, acting vertically downward. This weight is uniformly distributed along the entire length of the plank.
3. Reaction force at the fulcrum: Since the plank is in equilibrium, there must be an upward reaction force at the fulcrum (the point where the plank rests on the edge of the roof). This force counteracts the combined weight of the man and the plank.
Now, let's calculate the distance from the edge of the roof where the plank can overhang:
1. Determine the total weight acting on the plank: The total weight is the sum of the man's weight (200 lb) and the plank's weight (100 lb). So, the total weight is 300 lb.
2. Calculate the moment produced by the total weight: Since the plank is in equilibrium, the clockwise moment produced by the man's weight and the counterclockwise moment produced by the plank's weight must balance each other. The moment of an object is calculated by multiplying its weight by the distance from the fulcrum. Let's assume the distance from the fulcrum to the end of the plank where the man stands is "x" ft.
The moment produced by the man's weight = 200 lb * x ft
The moment produced by the plank's weight = 100 lb * (20 ft - x ft)
Since the moments are balanced, we can set them equal to each other:
200x = 100(20 - x)
3. Solve the equation to find the value of x: Distribute and simplify the equation:
200x = 2000 - 100x
300x = 2000
x = 2000 / 300
x ≈ 6.67 ft
Therefore, the plank can overhang approximately 6.67 ft from the edge of the roof.