A street lamp weighs 158 N. It is supported
equally by two wires that form an angle of
107˚ with each other.
What is the tension of each of these wires?
To calculate the tension in each of the wires supporting the street lamp, we can use the equilibrium conditions.
Let's assume that T1 and T2 represent the tensions in the two wires.
The forces acting on the street lamp can be resolved into vertical and horizontal components.
Let's analyze the vertical forces first:
Sum of vertical forces = 0
The weight of the street lamp acts vertically downwards and can be resolved into two components:
- The vertical component (Wv) = weight * sin(θ)
- The horizontal component (Wh) = weight * cos(θ)
Since the lamp is supported equally by two wires, the vertical components of the tensions in each wire will balance the weight of the lamp.
So, T1 sin(θ) + T2 sin(θ) = weight * sin(θ)
Now, analyzing the horizontal forces:
Sum of horizontal forces = 0
The horizontal components of the tensions in the wires will balance each other out.
Therefore, T1 cos(θ) - T2 cos(θ) = 0
This means that T1 cos(θ) = T2 cos(θ)
Now, we can solve the equations simultaneously to find the values of T1 and T2.
T1 sin(θ) + T2 sin(θ) = weight * sin(θ)
Simplifying, we have:
2T sin(θ) = weight * sin(θ)
T sin(θ) = weight * sin(θ) / 2
T = weight / 2
Substituting in the values:
T = 158 N / 2
T = 79 N
Therefore, the tension in each of the wires supporting the street lamp is 79 N.
To find the tension in each wire, we can use trigonometry.
Let's call the tension in one of the wires T1 and the tension in the other wire T2.
We know that the total weight of the lamp is 158 N, and it is being supported equally by the two wires.
Since the wires form an angle of 107 degrees with each other, we can split the weight of the lamp into two components along each wire.
The component of the weight acting on T1 can be found by using the cosine of the angle between T1 and the vertical direction. Similarly, the component of the weight acting on T2 can be found using the cosine of the angle between T2 and the vertical direction.
So, the equation for T1 can be written as:
T1 * cos(107˚) = 158 N/2
Simplifying this equation, we get:
T1 = (158 N/2) / cos(107˚)
Similarly, the equation for T2 can be written as:
T2 * cos(107˚) = 158 N/2
Simplifying this equation, we get:
T2 = (158 N/2) / cos(107˚)
Now we can plug in these values into a calculator to find the tensions in each wire.