A uniform beam of length L = 7.80 m and weight 4.05 102 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. (Sam is 1m from one end of beam and joe is 2m from the opposite end.)
Points:
left end - A, Sam - B, center of mass - C , Joe - D , right end - E
Torques about the point B:
mg•BC –F(J) •BD =0
F(J) =mg•BC/BD= 405•2.9/4.8 = 244.7(N)
Torques about the point D
F(S) •BD - mg•CD = 0
F(S)= mg•CD/BD = 405•1.9/4.8=160.3 N
To determine the force that each person exerts on the beam, we can use the principle of moments. The principle states that in equilibrium, the sum of the clockwise moments is equal to the sum of the anticlockwise moments.
In this case, we can balance the moments about any point on the beam. Let's choose one of the workers, Sam, as the pivot point. Now, let's calculate the moments exerted by different forces on the beam.
Since the beam is uniform, we can consider its weight acting at its center of gravity, which is at a distance of L/2 = 7.80 m / 2 = 3.90 m from Sam. The weight of the beam creates a counterclockwise moment about the pivot point, which we'll denote as M1.
M1 = Weight of the beam x Distance of the center of gravity from Sam
= 4.05 x 10^2 N x 3.90 m
Next, we need to consider the force exerted by Sam, which is acting at a distance of 1 m from him. This force creates a clockwise moment about the pivot point, which we'll denote as M2.
M2 = Force exerted by Sam x Distance from Sam
= Force exerted by Sam x 1 m
Finally, we have the force exerted by Joe, which is acting at a distance of 2 m from Sam. This force also creates a clockwise moment about the pivot point, denoted as M3.
M3 = Force exerted by Joe x Distance from Sam
= Force exerted by Joe x 2 m
According to the principle of moments, the sum of the clockwise moments should be equal to the sum of the counterclockwise moments.
M2 + M3 = M1
Now, to solve for the force exerted by each person, we divide both sides of the equation by the respective distances.
(Force exerted by Sam x 1 m) + (Force exerted by Joe x 2 m) = (4.05 x 10^2 N x 3.90 m)
Simplifying the equation, we get:
Force exerted by Sam + 2 x Force exerted by Joe = 4.05 x 3.90 x 10^2 N
Now, we have one equation with two unknowns. However, we also know that the sum of the forces exerted by both workers should be equal to the weight of the beam.
Force exerted by Sam + Force exerted by Joe = 4.05 x 10^2 N
We can now solve these two equations simultaneously to find the forces exerted by Sam and Joe on the beam.