A uniform plank of length 5.6 m and weight 203 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 440 N walk on the overhanging part of the plank before it just begins to tip
Write the sum of moments equation about any point, set it to zero. Remember the sum of vertical forces is zero.
I will be happy to critique your thinking or work.
To solve this problem, let's consider the forces and moments acting on the plank.
First, let's establish the coordinate system. We'll take the left end of the plank as the origin (0 m) and measure distances to the right.
The forces acting on the plank are:
1. The weight of the plank (203 N) acting vertically downward at the center of the plank.
2. The weight of the person (440 N) acting downward at the overhanging end of the plank.
The supports exert forces on the plank:
3. The left support exerts an upward force on the plank at the left end.
4. The right support exerts an upward force on the plank at the right end.
To determine the distance x that the person can walk on the overhanging part of the plank before it tips, we need to find the point where the plank is just about to tip, i.e., when it is in rotational equilibrium.
By taking moments about any point (let's choose the left end as the point), we can write the sum of moments equation as follows:
Sum of moments = 0
The moment M due to the weight of the person can be calculated as M = weight_person * distance_x. Here, weight_person is 440 N, and distance_x is the variable we want to determine.
The moment due to the weight of the plank is acting at the center of the plank, which is (5.6 m / 2 - 1.1 m) = 1.7 m from the left end. Therefore, the moment due to the weight of the plank is given by M = weight_plank * distance_to_center_of_weight_plank.
Since the plank is in rotational equilibrium, the sum of moments is zero. So the equation becomes:
M_person - M_plank = 0
weight_person * distance_x - weight_plank * distance_to_center_of_weight_plank = 0
Now, we can substitute the given values into our equation:
440 N * distance_x - 203 N * 1.7 m = 0
Solving for distance_x gives:
distance_x = (203 N * 1.7 m) / 440 N
distance_x ≈ 0.785 m
Therefore, a person who weighs 440 N can walk approximately 0.785 m on the overhanging part of the plank before it starts to tip.