Answer:
To solve the problem, we use the equations of equilibrium to find the tension in each cable holding up the loudspeaker. Since the loudspeaker is in equilibrium, the sum of the forces acting on it is zero. The weight of the loudspeaker is calculated first, and then we use trigonometry to find the horizontal and vertical components of the tension in one of the cables. We then apply the equation of equilibrium in the y direction to find the tension in each cable. The final answer is that the tension in each cable is approximately 81.1 N, which balances the weight of the loudspeaker.
___
Given:
Mass of loudspeaker (m) = 15.0 kg
Distance from ceiling (h) = 1.00 m
Length of cable (l) = 2.70 m
Acceleration due to gravity (g) = 9.80 m/s^2
Weight of loudspeaker:
Fg = mg
Fg = (15.0 kg)(9.80 m/s^2)
Fg = 147 N
Horizontal and vertical components of tension in one cable:
sinθ = h/l
sinθ = 1.00 m / 2.70 m
θ = sin^(-1)(1.00/2.70)
θ ≈ 21.6°
T_x = T sinθ
T_y = T cosθ
where T is the tension in the cable.
Equation of equilibrium in y direction:
ΣF_y = 2T cosθ - Fg = 0
Solving for T:
2T cosθ = Fg
T = Fg / (2 cosθ)
Plugging in the values:
T = (147 N) / (2 cos(21.6°))
T ≈ 81.1 N