A 1400-N uniform boom at ϕ = 70.0° to the horizontal is supported by a cable at an angle θ = 20.0° to the horizontal as shown in the figure below. The boom is pivoted at the bottom, and an object of weight w = 2250 N hangs from its top.
(a) Find the tension in the support cable.
kN
(b) Find the components of the reaction force exerted by the pivot on the boom.
horizontal component
kN right
vertical component
kN upward
Ah that "figure below" clause. Always sinks these questions.
To solve this problem, we can break down the forces acting on the boom and use trigonometry to find the required values.
Let's start with finding the tension in the support cable.
(a) Tension in the support cable:
Since the boom is in equilibrium, the sum of the vertical forces acting on it must be zero. This means that the vertical component of tension in the cable is equal in magnitude but opposite in direction to the weight of the object hanging from the boom.
Using trigonometry, we can calculate the tension in the cable as follows:
Tension = Weight / cos(θ)
Tension = 2250 N / cos(20°)
Let's calculate the value:
Tension = 2250 N / cos(20°) = 2358.34 N
Therefore, the tension in the support cable is approximately 2358.34 N.
(b) Components of the reaction force exerted by the pivot on the boom:
To find the components of the reaction force exerted by the pivot on the boom, we need to break down the force into vertical and horizontal components.
Let's start with the horizontal component:
The horizontal component of the reaction force exerted by the pivot is equal in magnitude and opposite in direction to the horizontal component of the tension in the cable.
Using trigonometry again, we can calculate the horizontal component as follows:
Horizontal Component = Tension * sin(θ)
Horizontal Component = 2358.34 N * sin(20°)
Let's calculate the value:
Horizontal Component = 2358.34 N * sin(20°) = 800.16 N
Therefore, the horizontal component of the reaction force exerted by the pivot on the boom is approximately 800.16 N to the right.
Now let's find the vertical component:
The vertical component of the reaction force exerted by the pivot is equal in magnitude and opposite in direction to the vertical component of the tension in the cable plus the weight of the boom.
Using trigonometry one more time, we can calculate the vertical component as follows:
Vertical Component = Tension * cos(θ) + Weight
Vertical Component = 2358.34 N * cos(20°) + 1400 N
Let's calculate the value:
Vertical Component = 2358.34 N * cos(20°) + 1400 N = 3731.19 N
Therefore, the vertical component of the reaction force exerted by the pivot on the boom is approximately 3731.19 N upward.
To summarize:
(a) The tension in the support cable is approximately 2358.34 N.
(b) The horizontal component of the reaction force exerted by the pivot is approximately 800.16 N to the right, and the vertical component is approximately 3731.19 N upward.
To find the tension in the support cable (part a), we can use trigonometry.
Since the boom is uniform, we can assume that the weight of the boom is acting at its center of mass, which is at the middle of the boom. The weight of the boom acts downward with a force of 1400 N.
To find the tension in the support cable, we need to resolve the forces acting on the boom.
First, let's resolve the weight of the boom into horizontal and vertical components. The weight of the boom has both horizontal and vertical components.
The vertical component is given by:
Vertical Component of Weight = Weight * sin(ϕ)
Vertical Component of Weight = 1400 N * sin(70.0°)
The horizontal component is given by:
Horizontal Component of Weight = Weight * cos(ϕ)
Horizontal Component of Weight = 1400 N * cos(70.0°)
Now, let's resolve the tension in the support cable into horizontal and vertical components. The tension in the cable has both horizontal and vertical components.
The vertical component is given by:
Vertical Component of Tension = Tension * sin(θ)
Vertical Component of Tension = Tension * sin(20.0°)
The horizontal component is given by:
Horizontal Component of Tension = Tension * cos(θ)
Horizontal Component of Tension = Tension * cos(20.0°)
Next, let's consider the vertical equilibrium of the boom. The vertical components of the weight and the tension in the support cable should add up to balance each other out.
Vertical Component of Weight + Vertical Component of Tension = 2250 N
Now, consider the horizontal equilibrium of the boom. The horizontal components of the weight and the tension in the support cable should cancel each other out since there is no acceleration in the horizontal direction.
Horizontal Component of Weight = Horizontal Component of Tension
Now, we can solve for the tension in the support cable. By substituting the values we have obtained:
1400 N * sin(70.0°) + Tension * sin(20.0°) = 2250 N
1400 N * cos(70.0°) = Tension * cos(20.0°)
By solving these equations simultaneously, we can find the tension in the support cable.
To find the components of the reaction force exerted by the pivot on the boom (part b), we can use the fact that the sum of the vertical and horizontal components of all the forces acting on an object must be equal to zero for equilibrium.
Since the boom is in equilibrium, the sum of the vertical components and horizontal components of the weight, the tension in the support cable, and the reaction force exerted by the pivot should add up to zero.
Let's call the horizontal component of the reaction force exerted by the pivot on the boom Rx and the vertical component Ry.
For vertical equilibrium:
Vertical Component of Weight + Vertical Component of Tension + Ry = 0
For horizontal equilibrium:
Horizontal Component of Weight + Horizontal Component of Tension + Rx = 0
By substituting the values we obtained earlier for the vertical component of weight and tension, we can find the vertical component of the reaction force exerted by the pivot on the boom.
Similarly, by substituting the values we obtained earlier for the horizontal component of weight and tension, we can find the horizontal component of the reaction force exerted by the pivot on the boom.