A traffic light weighting 145 newtons hangs from a vertical cable tied to two other cables that are fastened to a support. The upper cables make angles of 31.0 degrees and 57.0 degrees with the horizontal. Find the tension in each of the three cables.
To solve this problem, we can break down the forces acting on the traffic light.
First, let's define the tensions in the three cables:
T₁ - tension in the cable making an angle of 31.0 degrees with the horizontal
T₂ - tension in the cable making an angle of 57.0 degrees with the horizontal
T₃ - tension in the vertical cable
Now, let's analyze the forces acting on the traffic light:
1. The weight of the traffic light (145 N) acts vertically downward. We can represent this force as a vector pointing downwards.
2. The tension in the vertical cable (T₃) acts in the opposite direction to balance the weight of the traffic light. We can represent this force as a vector pointing upwards.
3. The tension in the horizontal cable (T₁) with an angle of 31.0 degrees acts towards the right to balance the horizontal component of the tension in the vertical cable. We can represent this force as a vector pointing to the right.
4. The tension in the horizontal cable (T₂) with an angle of 57.0 degrees acts towards the left to balance the horizontal component of the tension in the vertical cable. We can represent this force as a vector pointing to the left.
Since the system is in equilibrium, the vector sum of these forces must be zero. This means that the upward and downward forces must cancel each other out, and the horizontal forces must balance as well.
Now, let's use trigonometry to find the horizontal and vertical components of the forces:
For the tension T₁:
Horizontal component = T₁ * cos(31.0°)
Vertical component = T₁ * sin(31.0°)
For the tension T₂:
Horizontal component = T₂ * cos(57.0°)
Vertical component = T₂ * sin(57.0°)
Since the vertical components of T₁ and T₂ must cancel out the weight of the traffic light, we have:
T₁ * sin(31.0°) + T₂ * sin(57.0°) = 145 N
And since the horizontal components of T₁ and T₂ must balance each other, we have:
T₁ * cos(31.0°) = T₂ * cos(57.0°)
Now we can solve this system of equations to find the tensions in each cable.