In Figure 12-74, a uniform beam of length 11.7 m is supported by a horizontal cable and a hinge at angle θ = 52.7°. The tension in the cable is 444 N. What are (a) the x-component and (b) the y-component of the gravitational force on the beam? What are (c) the x-component and (d) the y-component of the force on the beam from the hinge?
To solve this problem, we can break down the forces acting on the beam into their x and y components. Let's go step by step:
(a) The x-component of the gravitational force on the beam:
To find the x-component of the gravitational force, you need to consider the angle at which the beam is inclined. Assuming that the positive x-direction is horizontal and to the right, the x-component of the gravitational force will be pointing downwards and to the right.
We can find the x-component of the gravitational force using trigonometry. Given the angle θ = 52.7° and the total gravitational force Fg, the x-component of the gravitational force Fgx can be determined using the equation:
Fgx = Fg * cos(θ)
Here, Fg is the total gravitational force acting on the beam. In this problem, we need to calculate the x-component of the gravitational force, so we will need to find Fg first.
The total gravitational force acting on the beam can be calculated using the equation:
Fg = m * g
Where m is the mass of the beam and g is the acceleration due to gravity (approximately 9.8 m/s^2).
(b) The y-component of the gravitational force on the beam:
Similarly, we can find the y-component of the gravitational force using trigonometry. The y-component of the gravitational force, Fgy, will be pointing downwards and vertically downwards.
The equation to find the y-component of the gravitational force is:
Fgy = Fg * sin(θ)
(c) The x-component of the force on the beam from the hinge:
To find the x-component of the force on the beam from the hinge, we need to consider the equilibrium of forces. The x-component of the force from the hinge will be equal in magnitude but opposite in direction to the x-component of the tension in the cable.
Since the beam is in equilibrium, the net force in the x-direction on the beam is zero. Therefore, the x-component of the force from the hinge will be equal in magnitude but opposite in direction to the x-component of the tension in the cable.
So, the x-component of the force on the beam from the hinge will be:
Fhx = -Tension_x
Here, Tension_x is the x-component of the tension in the cable.
(d) The y-component of the force on the beam from the hinge:
Similarly, the y-component of the force from the hinge will be equal in magnitude but opposite in direction to the y-component of the tension in the cable.
The y-component of the force on the beam from the hinge will be:
Fhy = -Tension_y
Here, Tension_y is the y-component of the tension in the cable.
By finding the tension components first and then using these components to calculate the hinge force components, we can determine the x and y components of each force acting on the beam.