A 1200-N uniform beam is attached to a vertical wall at one end and is supported by a cable at the other end. A W = 1825-N crate hangs from the far end of the beam.
(a) Using the data shown in the drawing, find the magnitude of the tension in the wire.
(b) Using the data shown in the drawing, find the magnitude of the horizontal and vertical components of the force that the wall exerts on the left end of the beam.
To solve this problem, we can apply the principles of equilibrium. In equilibrium, the net force and net torque must both be zero.
(a) To find the magnitude of the tension in the wire, we need to consider the vertical equilibrium of the beam. Since the beam is in equilibrium, the sum of the vertical forces acting on it must be zero.
We have two vertical forces acting on the beam: the weight of the crate and the tension in the wire. The weight of the crate is given as W = 1825 N. Since the weight acts downwards, it is considered negative.
Using the principle of vertical equilibrium, we can write the equation:
-Sum of upward forces = Sum of downward forces
The only upward force is the tension in the wire, so the equation becomes:
Tension - Weight of crate = 0
Therefore, the magnitude of the tension in the wire is equal to the weight of the crate:
Magnitude of tension = |Weight of crate| = |1825 N| = 1825 N.
So, the magnitude of the tension in the wire is 1825 N.
(b) To find the magnitude of the horizontal and vertical components of the force that the wall exerts on the left end of the beam, we need to consider the horizontal and vertical equilibrium of the beam.
In horizontal equilibrium, the sum of the horizontal forces acting on the beam must be zero. Since there are no horizontal forces acting on the beam, the horizontal component of the force exerted by the wall is zero.
In vertical equilibrium, the sum of the vertical forces acting on the beam must be zero. We have two vertical forces acting on the beam, the weight of the beam itself and the vertical component of the force exerted by the wall.
Since the weight of the beam acts downward, it is considered negative. Therefore, using the principle of vertical equilibrium, we can write the equation:
-Sum of upward forces = Sum of downward forces
The upward force is the vertical component of the force exerted by the wall, so the equation becomes:
Vertical component of force exerted by wall - Weight of beam = 0
Therefore, the magnitude of the vertical component of the force exerted by the wall is equal to the weight of the beam:
Magnitude of vertical component of force = |Weight of beam| = |1200 N| = 1200 N.
So, the magnitude of the horizontal component of the force that the wall exerts on the left end of the beam is zero, and the magnitude of the vertical component of the force is 1200 N.