A uniform steel beam of length L and mass m 1 is attached via a hinge to the sid

Question

A uniform steel beam of length L and mass m1 is attached via a hinge to the side of a building. The beam is supported by a steel cable attached to the end of the beam at an angle ?, as shown. Through the hinge, the wall exerts an unknown force, F, on the beam. A workman of mass m2 sits eating lunch a distance d from the building. (Figure 1)

Part A

Find T, the tension in the cable. Remember to account for all the forces in the problem.

Express your answer in terms of m1, m2, L, d, ?, and g, the magnitude of the acceleration due to gravity.

Part B

Find Fx, the x-component of the force exerted by the wall on the beam ( F), using the axis shown. Remember to pay attention to the direction that the wall exerts the force.

Express your answer in terms of T and other given quantities.

Explanation / Answer

A) let T is the tension in the string.

As the beam is in equilibrium, net force and net torque acting on the beam must be zero.

Apply net torque about hinge = 0

T*L*sin(theta) - m1*g*(L/2) - m2*g*d = 0

T*L*sin(theta) = m1*g*(L/2) + m2*g*d

T = (m1*g*(L/2) + m2*g*d)/(L*sin(theta))

= (m1*g*L + 2*m2*g*d)/(2*L*sin(theta))

B)
Apply, Fnetx = 0

T*cos(theta) - Fx = 0

Fx = T*cos(theta)

= (m1*g*(L/2) + m2*g*d)*cos(theta)/(L*sin(theta))

= (m1*g*(L/2) + m2*g*d)/(L*tan(theta))

= (m1*g*L + 2*m2*g*d)/(2*L*tan(theta))

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