a uniform beam with a weight of 51.3 N and a length of 3.23 m is hinged at its lower end, and a horizontal force Upper F Overscript right-arrow EndScripts of magnitude 64.0 N acts at its upper end. The beam is held vertical by a cable that makes angle θ = 20.7° with the ground and is attached to the beam at height h = 2.29 m. What are (a) the tension in the cable, (b) the x-component of the force on the beam from the hinge, and (c) the y-component of the force on the beam from the hinge.
To solve this problem, we can use the principles of equilibrium.
Step 1: Draw a Free Body Diagram (FBD) of the beam:
- Draw the beam as a straight line and label its length as 3.23 m.
- Label the weight of the beam as 51.3 N.
- Draw a vertical arrow at the lower end of the beam representing the force vector due to the hinge. (This is the force exerted by the hinge on the beam.)
- Draw a horizontal arrow at the upper end of the beam representing the force vector due to the applied force. (This is the force exerted by the applied force on the beam.)
- Draw a diagonal arrow representing the tension in the cable attached to the beam.
Step 2: Analyze forces acting on the beam:
- The weight of the beam acts vertically downward, so its force vector points downward, with a magnitude of 51.3 N.
- The force on the beam due to the applied force is horizontal and has a magnitude of 64.0 N.
- The cable exerts a force on the beam, which can be divided into horizontal and vertical components.
- The vertical component of the cable force acts upward and is equal to the tension in the cable.
- The horizontal component of the cable force is equal to the force exerted by the hinge on the beam.
Step 3: Write equations of equilibrium:
- For vertical equilibrium, the sum of the vertical forces acting on the beam must be zero. This gives us the equation:
Tension in the cable - Weight of the beam = 0
Tension in the cable = Weight of the beam
Tension in the cable = 51.3 N
- For horizontal equilibrium, the sum of the horizontal forces acting on the beam must be zero. This gives us the equation:
Horizontal component of the cable force + Force due to applied force - Horizontal force from the hinge = 0
Horizontal component of the cable force + 64.0 N - Horizontal force from the hinge = 0
Horizontal component of the cable force = Horizontal force from the hinge - 64.0 N
- The horizontal component of the cable force is equal to the force exerted by the hinge on the beam.
Step 4: Calculate the horizontal force from the hinge:
- To calculate the horizontal force from the hinge, we need to find the horizontal component of the tension in the cable. This can be done using trigonometry.
Cos(θ) = adjacent/hypotenuse
Cos(20.7°) = Horizontal component of the tension in the cable / Tension in the cable
Rearranging the equation:
Horizontal component of the tension in the cable = Tension in the cable * Cos(20.7°)
Plugging in the values:
Horizontal component of the tension in the cable = 51.3 N * Cos(20.7°)
Step 5: Calculate the y-component of the force on the beam from the hinge:
- The y-component of the force on the beam from the hinge can be found using trigonometry.
Sin(θ) = opposite/hypotenuse
Sin(20.7°) = y-component of the force on the beam from the hinge / Tension in the cable
Rearranging the equation:
y-component of the force on the beam from the hinge = Tension in the cable * Sin(20.7°)
Plugging in the values:
y-component of the force on the beam from the hinge = 51.3 N * Sin(20.7°)
Step 6: Calculate the results:
(a) The tension in the cable is 51.3 N.
(b) The x-component of the force on the beam from the hinge is 51.3 N * Cos(20.7°).
(c) The y-component of the force on the beam from the hinge is 51.3 N * Sin(20.7°).
To solve this problem, we can break it down into components and analyze the forces acting on the beam.
First, let's find the tension in the cable (a):
The vertical force component exerted by the cable can be determined using the equation:
Tension = Weight of the Beam + Vertical Component of the Applied Force – Vertical Component of the Hinge Force
1. Weight of the beam: Given that the weight of the beam is 51.3 N.
2. Vertical component of the applied force: The applied force is horizontal, and we need to find its vertical component. We can calculate it using the equation:
Vertical Component of Applied Force = Applied Force * sin(θ)
where θ is the angle made by the cable with the ground, given as 20.7°, and the applied force is 64.0 N.
3. Vertical component of the hinge force: The hinge force also has a vertical component as the beam is held vertical. We can calculate it using the equation:
Vertical Component of Hinge Force = Hinge Force * cos(θ)
where θ is the same angle as before, which is 20.7°, and the hinge force is the force exerted by the hinge to keep the beam vertical.
Now, we can calculate the tension in the cable:
Tension = 51.3 N + (64.0 N * sin(20.7°)) - (Hinge Force * cos(20.7°))
Next, let's find the x-component of the force on the beam from the hinge (b):
Since the beam is held vertical, the x-component of the force from the hinge is equal to zero because there is no horizontal force acting on it.
Therefore, the x-component of the force on the beam from the hinge is 0 N.
Finally, let's find the y-component of the force on the beam from the hinge (c):
To find the y-component of the force, we can use the equation:
Y-Component of Hinge Force = Hinge Force * sin(θ)
where θ is the same angle as before, which is 20.7°, and the hinge force is the force exerted by the hinge to keep the beam vertical.
Now, let's calculate the Y-Component of the Hinge Force using this equation.