Question

A chandelier with mass m is attached to the ceiling of a large concert hall by two cables. Because the ceiling is covered with intricate architectural decorations (not indicated in the figure, which uses a humbler depiction), the workers who hung the chandelier couldn't attach the cables to the ceiling directly above the chandelier. Instead, they attached the cables to the ceiling near the walls. Cable 1 has tension T1 and makes an angle of θ1 with the ceiling. Cable 2 has tension T2 and makes an angle of θ2 with the ceiling.

Answer 1

Take components of the tensions. With angles measured from the ceiling (i.e. from the horizontal), equilibrium gives

- Horizontal: T1 cosθ1 = T2 cosθ2
- Vertical: T1 sinθ1 + T2 sinθ2 = mg

Solve the two equations. From the horizontal equation T1 = T2 cosθ2/cosθ1. Substituting into the vertical equation and simplifying (use cosθ2 sinθ1 + sinθ2 cosθ1 = sin(θ1+θ2)) gives

T2 = mg cosθ1 / sin(θ1 + θ2),
T1 = mg cosθ2 / sin(θ1 + θ2).

Special case: if θ1 = θ2 = θ, then T1 = T2 = mg/(2 sinθ).