Question

(Figure 1)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.

Question
Find an expression for T1, the tension in cable 1, that does not depend on T2. Express your answer in terms of some or all of the variables m, θ1, and θ2, as well as the magnitude of the acceleration due to gravity g. You must use parentheses around θ1 and θ2, when they are used as arguments to any trigonometric functions in your answer.

Answer 1

To find the expression for T1, the tension in cable 1, we can use trigonometry and the equilibrium condition. The expression is T1 = (mg - T2*sin(theta2)) / sin(theta1).

Step-by-step explanation:

To find an expression for T1, the tension in cable 1, we can use trigonometry and the equilibrium condition. The forces acting on the chandelier are its weight mg and the tensions T1 and T2 in the cables. Let's consider the vertical forces. The vertical component of T1 is T1*sin(theta1) and the vertical component of T2 is T2*sin(theta2). Since the chandelier is not accelerating vertically, the sum of the vertical components of the tensions must equal the weight mg. This can be expressed as:

T1*sin(theta1) + T2*sin(theta2) = mg

Solving for T1:

T1 = (mg - T2*sin(theta2)) / sin(theta1)