a light fixtures of 80.o N hangs from a verttical chain that is tied to two other cables that are fastened to support. Both the upper cables make an angle of 37.0 with respect to the horizontial. calculate the tension in the left and right cables
The sum of the horizontal tension components is zeo and the sum of the vertical tension components equals the weight. The two tension forces are also equal in magnitude, due to symmetry.
To calculate the tension in the left and right cables, we can break down the forces acting on the light fixture and solve for the unknown tensions.
Let's define the following variables:
T₁ = Tension in the left cable
T₂ = Tension in the right cable
First, let's analyze the forces acting vertically on the light fixture:
T₁ * sin(37.0°) - T₂ * sin(37.0°) - 80.0 N = 0
(Sum of upward forces) - (Sum of downward forces) - (Weight of the light fixture) = 0
Now, let's analyze the forces acting horizontally on the light fixture:
T₁ * cos(37.0°) + T₂ * cos(37.0°) = 0
(Sum of forces towards the right) + (Sum of forces towards the left) = 0
Using these two equations, we can solve for T₁ and T₂.
Here's the step-by-step solution:
1. Substitute the values into the equations:
T₁ * sin(37.0°) - T₂ * sin(37.0°) - 80.0 N = 0
T₁ * cos(37.0°) + T₂ * cos(37.0°) = 0
2. Simplify the equations:
0.6T₁ - 0.6T₂ - 80.0 = 0
0.8T₁ + 0.8T₂ = 0
3. Solve the second equation for T₁:
T₁ = -T₂
4. Replace T₁ in the first equation with -T₂:
0.6(-T₂) - 0.6T₂ - 80.0 = 0
5. Simplify and solve for T₂:
-1.2T₂ - 80.0 = 0
-1.2T₂ = 80.0
T₂ = 80.0 / -1.2
T₂ ≈ -66.7 N (rounded to one decimal place)
6. Substitute T₂ = -66.7 N back into the second equation to find T₁:
0.8T₁ + 0.8(-66.7) = 0
0.8T₁ - 53.4 = 0
0.8T₁ = 53.4
T₁ = 53.4 / 0.8
T₁ ≈ 66.7 N (rounded to one decimal place)
Therefore, the tension in the left cable (T₁) is approximately 66.7 N, and the tension in the right cable (T₂) is approximately -66.7 N.