In Fig. 12-51, uniform beams A and B are attached to a wall with hinges and loosely bolted together. Beam A has length LA = 2.55 m and mass 49.0 kg; beam B has mass 69.0 kg. The two hinge points are separated by distance d = 1.86 m. In unit-vector notation, what is the force on (a) beam A due to its hinge, (b) beam A due to the bolt, (c) beam B due to its hinge, and (d) beam B due to the bolt?
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To solve this problem, we can use Newton's second law and the principles of equilibrium. Let's go through each part of the problem step by step:
(a) Force on beam A due to its hinge:
We can start by considering the forces acting on beam A. Since it is attached to the wall with a hinge, the only external force acting on it is the force of gravity. The force of gravity can be broken down into two components: one perpendicular to the beam (normal force) and one parallel to the beam (weight).
The normal force acts along the direction of the hinge and acts as a centripetal force to keep the beam in circular motion. Since the beam is in equilibrium, the sum of the forces acting on it should be zero. The force exerted by the hinge can be represented using unit-vector notation as:
F_hingeA = -m_A * g * n_A
Here, m_A is the mass of beam A, g is the acceleration due to gravity, and n_A is the unit vector along the hinge direction.
(b) Force on beam A due to the bolt:
The bolt connecting beams A and B exerts a force on beam A. This force will be directed along the line of action of the bolt, which is perpendicular to the beam. To find the force exerted by the bolt, we need to consider the torque equilibrium about the hinge point of beam A. Since the beam is at equilibrium, the torque exerted by the bolt and the weight of beam A should balance each other. We can represent the force exerted by the bolt using unit-vector notation as:
F_boltA = (m_A * g * d / LA) * t_hinge
Here, t_hinge is the unit vector perpendicular to the hinge direction.
(c) Force on beam B due to its hinge:
Similar to beam A, beam B is also attached to the wall with a hinge. Therefore, the only external force acting on beam B is the force of gravity. The force exerted by the hinge on beam B can be represented using unit-vector notation as:
F_hingeB = -m_B * g * n_B
Here, m_B is the mass of beam B, g is the acceleration due to gravity, and n_B is the unit vector along the hinge direction.
(d) Force on beam B due to the bolt:
The bolt connecting beams A and B exerts a force on beam B. Since beam B is in equilibrium, the torque exerted by the bolt and the weight of beam B should balance each other. The force exerted by the bolt can be represented using unit-vector notation as:
F_boltB = (m_B * g * d / LB) * t_hinge
Here, LB is the length of beam B and t_hinge is the unit vector perpendicular to the hinge direction.
By using these equations, you can calculate the force on beam A due to its hinge, the force on beam A due to the bolt, the force on beam B due to its hinge, and the force on beam B due to the bolt using the given values for mass, length, and distance.