Ten bags of top soil, each weighing 180 N, are placed on a 2.90-m long sheet of wood. They are stacked 0.50 m from one end of the sheet of wood, as shown below. Two people lift the sheet of wood, one at each end. Ignore the weight of the wood, how much force must each person exert?
To find out how much force each person must exert to lift the sheet of wood with the ten bags of topsoil, we need to consider the weight of the bags and the distribution of that weight along the sheet of wood.
First, let's find the total weight of the bags. We have ten bags, each weighing 180 N. Therefore, the total weight is 10 bags * 180 N/bag = 1800 N.
Next, we need to consider the distribution of weight along the sheet of wood. The bags are stacked 0.50 m from one end, which means that the distance between the stack of bags and the person at one end is 0.50 m. The distance between the stack of bags and the person at the other end is 2.90 m - 0.50 m = 2.40 m.
To determine the force that each person must exert, we can use the principle of moments, which states that the total moment on an object is equal to the sum of the moments of all the forces acting on it. In this case, we can set up the equation:
Force1 * Distance1 = Force2 * Distance2
Where:
Force1 is the force exerted by the person at one end,
Distance1 is the distance between the person at one end and the stack of bags (0.50 m),
Force2 is the force exerted by the person at the other end,
Distance2 is the distance between the person at the other end and the stack of bags (2.40 m).
Since the total weight of the bags is 1800 N, the total force exerted by the two people combined must be 1800 N to balance it. Therefore, we have:
Force1 + Force2 = 1800 N
Substituting the values for the distances, we have:
Force1 * 0.50 m = Force2 * 2.40 m
Now we can solve these two equations simultaneously to find the forces exerted by each person:
Force1 + Force2 = 1800 N (equation 1)
Force1 * 0.50 m = Force2 * 2.40 m (equation 2)
From equation 2, we can rearrange it to get Force1 in terms of Force2:
Force1 = (Force2 * 2.40 m) / 0.50 m = 4.80 * Force2
Substituting this expression for Force1 in equation 1, we have:
4.80 * Force2 + Force2 = 1800 N
Combining like terms, we get:
5.80 * Force2 = 1800 N
Now we can solve for Force2:
Force2 = 1800 N / 5.80 = 310.34 N
Finally, we can substitute this value back into the equation to find Force1:
Force1 = 4.80 * Force2 = 4.80 * 310.34 N = 1484.15 N
Therefore, each person must exert approximately 310.34 N and 1484.15 N of force, respectively, to lift the sheet of wood with the ten bags of topsoil.