Review Conceptual Example 7 before starting this problem. A uniform plank of length 5.0 m and weight 225 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 451 N walk on the overhanging part of the plank before it just begins to tip?
If the right support is 1.1m from the end, then 3.9m, or 78% hangs over the left. 78% of 225N is exerted downward from the 'center' of the left side of the board. That 'center' works out to be half of 3.9m, or 1.95m.
On the right side, you have 22% of the board. 22% of 225N acts downward 0.55m away (half of 1.1m) from the support. Then you have a person of 451N on the right side, as well, an unknown away from the support. You need to equal out all of the forces times distances.
(0.78 • 225) •1.95 = ((0.22 • 225) •0.55) + (451•X )
Solve for X.
To solve this problem, we need to consider the tipping point of the plank. The plank will begin to tip when the torque created by the person's weight overcomes the torque created by the weight of the plank itself.
Let's break down the problem step by step:
1. Start by calculating the torque created by the person's weight. Torque is calculated using the formula:
Torque = Force x Distance
In this case, the force is the weight of the person, which is given as 451 N, and the distance is the distance x we want to find.
So the torque created by the person's weight is: Torque_person = 451 N x x
2. Next, calculate the torque created by the weight of the plank. Torque is again calculated using the formula:
Torque = Force x Distance
The force is the weight of the plank, which is given as 225 N, and the distance is the overhang of the plank, which is given as 1.1 m.
So the torque created by the weight of the plank is: Torque_plank = 225 N x 1.1 m
3. The plank will begin to tip when the torque created by the person's weight is equal to the torque created by the weight of the plank. Therefore, we can set up the following equation:
Torque_person = Torque_plank
451 N x x = 225 N x 1.1 m
4. Solve the equation for x. Start by dividing both sides of the equation by 451 N:
x = (225 N x 1.1 m) / 451 N
Simplify the equation:
x = 0.544 m
Therefore, a person who weighs 451 N can walk on the overhanging part of the plank for a maximum distance of 0.544 m before it just begins to tip.