Final answer:
To find the tensions in the wires supporting the spotlight, we use the principles of static equilibrium to set up two equations: one for the vertical forces and one for the horizontal forces. Solving these simultaneously with the given angles and spotlight weight will yield the tensions for each wire.
Step-by-step explanation:
To find the tension in each wire supporting a 75 kilogram spotlight with one wire at a 13° angle to the horizontal and the other at a 75° angle to the vertical, we need to apply the principles of static equilibrium. In static equilibrium, forces in the x-direction (horizontal) and y-direction (vertical) must sum to zero because the spotlight is at rest.
First, we analyze the forces in the vertical direction. The entire weight of the spotlight, which is the force due to gravity (Fg = m*g), must be supported by the vertical components of the tensions in both wires (T1 and T2). If we let T1y and T2y represent the vertical components of the tensions, then:
- Fg = T1y + T2y
- Fg = T1*sin(θ_1) + T2*sin(θ_2)
Where θ_1 is the angle the left wire makes with the horizontal (77°, since it's 13° off from the vertical) and θ_2 is the angle the right wire makes with the horizontal (75°).
Next, the horizontal forces must cancel each other out:
- T1*cos(θ_1) = T2*cos(θ_2)
We have two equations and two unknowns (T1 and T2), which we can solve simultaneously to find the tensions in the wires. By substituting the known values (m = 75 kg, g = 9.8 m/s², θ_1 = 77°, and θ_2 = 75°) into the equations and solving them, we can determine the tensions that balance the forces in each direction.