A uniform beam of length L = 8.00 m and weight 3.80 102 N is carried by two workers, Sam and Joe, as shown in the figure below. Determine the force that each person exerts on the beam.
x10^2N*
To determine the force that each person exerts on the beam, we need to consider the equilibrium condition. In equilibrium, the sum of all the forces acting on the beam must be zero.
Given:
Length of the beam (L) = 8.00 m
Weight of the beam (W) = 3.80 x 10^2 N
Let's analyze the forces acting on the beam:
1. Weight of the beam: This force acts vertically downward, towards the center of the Earth. Its magnitude is equal to the weight of the beam, which is given as 3.80 x 10^2 N.
2. Force exerted by Sam: Let's assume Sam exerts a force (F_s) on the beam at a distance x from the left end, as shown in the figure. This force acts vertically upward.
3. Force exerted by Joe: Let's assume Joe exerts a force (F_j) on the beam at a distance (L - x) from the left end. This force acts vertically upward.
Now, applying the equilibrium condition, we can write the equation for the sum of the forces in the vertical direction:
F_s + F_j - W = 0
Since the beam is uniform, we can assume that the weight of the beam acts at its center, which means the weight acts at a distance L/2 from the left end.
Now, substituting the known values into the equation:
F_s + F_j - 3.80 x 10^2 N = 0
To find the forces F_s and F_j, we can use the condition that the beam is in equilibrium. The beam will be in equilibrium if the sum of the torques (moments) about any point is zero.
Taking moments about the left end of the beam (point A in the figure), we have:
F_j x (L - x) - (3.80 x 10^2 N) x (L/2) = 0
Simplifying the equation:
F_j(L - x) - (1.90 x 10^2 N) x L = 0
F_jL - F_jx - 1.90 x 10^2 N x L = 0
Now, let's rearrange the equation to solve for F_j:
F_jL - F_jx = 1.90 x 10^2 N x L
Factor out F_j:
F_j(L - x) = 1.90 x 10^2 N x L
Divide both sides by (L - x):
F_j = (1.90 x 10^2 N x L) / (L - x)
Similarly, for F_s:
F_s = (1.90 x 10^2 N x L) / x
Now, substitute the value of L (8.00 m) into the equations to calculate the forces F_j and F_s.