A uniform plank of length 5.0 m and weight 217 N rests horizontally on two supports, with d = 1.06 m of the plank hanging over the right support. To what distance, x, can a person who weighs 427 N walk on the overhanging part of the plank before it just begins to tip?
Well, let's not tip over the answer here, but we can solve this! To find the tipping point, we need to make sure the moments are balanced.
First, let's calculate the weight of the plank. Since weight is equal to mass multiplied by gravity, and we know the weight is 217 N, we can divide that by the acceleration due to gravity (approximately 9.8 m/s^2) to find the mass. So, the mass of the plank is about 22.1 kg.
Now, we need to find the combined moment of the person and the plank. Since the person weighs 427 N, and they are walking at some distance x from the right support, the moment created by the person is 427 N multiplied by x.
For the plank to not tip, the total moment created by the person and the plank must be zero. This means the moment created by the person must be equal in magnitude but opposite in direction to the moment created by the plank.
So, 427 N multiplied by x must be equal to the moment created by the plank. The moment created by the plank can be calculated by multiplying the weight of the plank by the distance between the center of mass of the plank and the right support, which is (5.0 m - 1.06 m).
Setting these two moments equal to each other, we have:
427 N * x = 217 N * (5.0 m - 1.06 m)
Simplifying, we have:
427 N * x = 217 N * 3.94 m
Dividing both sides by 427 N, we find:
x = (217 N * 3.94 m) / 427 N
Calculating this, we find:
x ≈ 2.001 m
So, the person can walk approximately 2.001 meters on the overhanging part of the plank before it just begins to tip. Just be careful not to go overboard with your steps!
To determine the distance x that a person can walk on the overhanging part of the plank before it begins to tip, we need to calculate the moment produced by the person's weight and compare it to the moment produced by the weight of the plank.
First, let's find the weight of the overhanging part of the plank. Since the weight of the entire plank is 217 N, the weight of the overhanging part can be calculated by proportion:
Weight of overhanging part = (Length of overhanging part / Total length of plank) * Weight of entire plank
Weight of overhanging part = (1.06 m / 5.0 m) * 217 N
Now, let's calculate the moment produced by the person's weight when they walk on the overhanging part. The moment is given by the formula:
Moment = Force * Distance
Since the person weighs 427 N and the distance they walk is x, the moment produced by the person's weight is:
Moment from person = 427 N * x
Next, let's calculate the moment produced by the weight of the overhanging part of the plank. The weight acts at the center of the overhanging part, which is at a distance of (1.06 m / 2) = 0.53 m from the support. The moment is given by the formula:
Moment = Force * Distance
Since the weight of the overhanging part is calculated above and the distance from the support is 0.53 m, the moment produced by the weight of the overhanging part is:
Moment from overhanging part = Weight of overhanging part * 0.53 m
To avoid tipping, the moment produced by the person's weight should be equal to the moment produced by the weight of the overhanging part:
Moment from person = Moment from overhanging part
427 N * x = Weight of overhanging part * 0.53 m
Now we can solve for x:
x = (Weight of overhanging part * 0.53 m) / 427 N
Substitute the calculated value of the weight of the overhanging part and solve for x to find the maximum distance a person can walk on the overhanging part before it begins to tip.