A uniform plank of length 4.5 m and weight 233 N rests horizontally on two supports, with 1.1 m of the plank hanging over the right support (see the drawing). To what distance x can a person who weighs 454 N walk on the overhanging part of the plank before it just begins to tip?
The drawing would help understand where the left support is.
To determine the maximum distance x that a person can walk on the overhanging part of the plank before it begins to tip, we need to consider the moments (torques) acting on the plank.
In this scenario, there are two main moments to consider: the moment due to the weight of the plank itself and the moment due to the weight of the person.
1. Moment due to the weight of the plank:
The weight of the plank acts downward at its center, which is located at the midpoint of the entire length. Since the plank is uniform, the weight can be considered to act at this midpoint, which is 4.5 m / 2 = 2.25 m from the left support. Therefore, the moment due to the weight of the plank is:
Moment_plank = Weight_plank * Distance_plank
Weight_plank = 233 N
Distance_plank = 2.25 m
2. Moment due to the weight of the person:
The weight of the person acts vertically downward at any given position on the overhanging part of the plank. To find the maximum distance x, we need to determine the position on the overhanging part where the moment due to the person's weight is equal to the moment due to the plank's weight.
Moment_person = Weight_person * Distance_person
Weight_person = 454 N
Distance_person = 1.1 m + x (the distance person has walked)
The plank will just begin to tip when the two moments are equal:
Moment_plank = Moment_person
233 N * 2.25 m = 454 N * (1.1 m + x)
Now we can solve the equation to find the value of x:
524.25 N*m = 499.4 N*m + 454 N * x
24.85 N*m = 454 N * x
x = 24.85 N*m / 454 N
x ≈ 0.0547 m
Therefore, a person can walk approximately 0.0547 meters (or about 5.47 centimeters) on the overhanging part of the plank before it just begins to tip.