please show work A chandelier with mass m is attached to the ceiling of a large

Question

please show work

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 T_1 and makes an angle of with the ceiling. Cable 2 has tension T2 and makes an angle of theta_2 with the ceiling. Find an expression for T_1, the tension in cable 1, that does not depend on T2

Explanation / Answer

In the vertical direction, the forces are balanced:
Fnet = 0
T1sin1 + T2sin2 = mg

In the horizontal direction, the forces are also balanced:
Fnet = 0
T1cos1 = T2cos2
T2 = T1(cos1)/(cos2)

Using this result in the first equation:
T1sin1 + T2sin2 = mg
T1sin1 + T1(cos1)(sin2)/(cos2) = mg
T1sin1 + T1(cos1)(tan2) = mg
T1(sin1 + (cos1)(tan2)) = mg

T1 = mg cos2/(sin(1+2))

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