A 1245-N uniform beam is attached to a vertical wall at one end and is supported by a cable at the other end. A W = 1980-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.
N
(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.
horizontal
N
vertical
N
To solve this problem, we'll use the principle of equilibrium, which states that the sum of the forces acting on an object in equilibrium must be zero.
(a) To find the magnitude of the tension in the wire, we need to consider the forces acting on the beam. There are two vertical forces: the weight of the beam acting downward (1245 N) and the weight of the crate acting upward (1980 N).
Let's assume the tension in the wire is T. Since the beam is in equilibrium, the sum of the vertical forces must be zero:
T - 1245 N + 1980 N = 0
Simplifying the equation gives us:
T = 1245 N - 1980 N
T = -735 N
The magnitude of the tension in the wire is 735 N.
(b) To find the horizontal and vertical components of the force that the wall exerts on the left end of the beam, we need to consider the forces acting on the left end of the beam. There are two forces in play: the horizontal component of the tension in the wire (Tcosθ) and the vertical component of the tension in the wire (Tsinθ), where θ is the angle formed between the wall and the beam.
Since the beam is in equilibrium, the sum of the horizontal forces must be zero:
Tcosθ = 0
Therefore, the horizontal component of the force that the wall exerts on the left end of the beam is zero.
For the vertical component, we can use the fact that the sum of the vertical forces is zero:
Tsinθ - 1245 N + 1980 N = 0
Simplifying the equation gives us:
Tsinθ = 1245 N - 1980 N
Tsinθ = -735 N
Dividing both sides by T gives us:
sinθ = -735 N / T
From part (a), we found that T = 735 N. Substituting this value in the equation:
sinθ = -735 N / 735 N
sinθ = -1
Since sinθ = -1, we know that θ is 270 degrees.
Therefore, the magnitude of the horizontal component of the force that the wall exerts on the left end of the beam is 0 N, and the magnitude of the vertical component is -735 N.