Solving Problems with Newton's Laws - Hanging Chandelier

  1. 1. The problem statement, all variables and given/known data

    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.
    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.

    2. Relevant equations




    3. The attempt at a solution

    So far I have found the x and y components:

    ΣFx = 0 = -T1cosθ1 + T2cosθ2
    ΣFy = 0 = T1sinθ1 + T2sinθ2 - mg

    Now I need to eliminate T2 from this pair of equations and solve for T1, but I can't figure out how to do this.
     
  2. jcsd
  3. LowlyPion

    LowlyPion 5,335
    Homework Helper

    2 equations - 4 unknowns T1 and T2, θ1, θ2?

    You can't get rid of T2?

    Try harder. You know directly that T2 = T1*(Cosθ1/Cosθ2)

    Surely something will occur to you.
     
  4. I tried that, but I kept getting an answer where T1 was equal to something that had T1 in it. Such as:

    T1 = (mg - T1cosθ1tanθ2) / sinθ1

    I got that when I plugged T2 = T1 (cosθ1)/(cosθ2) into the y component.
     
  5. LowlyPion

    LowlyPion 5,335
    Homework Helper

    And this is a problem because ...?
     
  6. Is it allowed to have T1 in the answer? I didn't think it was because it said 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. It never mentions T1.
     
  7. T1 is not suppose t be in the answer. i have the same question for my assignment. the answer ends up being :
    mg/sin(theta1)+tan(theta2)*cos(theta1)

    i too substituted T2 = T1 (cosθ1)/(cosθ2) for T2 in the second equation and got

    0=T1sin(theta1)+T1(cos(theta1)*sin(theta2)/cos(theta2))-mg

    after moving the mg to the other side i made the sin(theta2)/cos(theta2)= tan(theta2), and then factored out the two.

    mg=T1(sin(theta1)+tan(theta2)cos(theta1)

    devided both sides by sin(theta1)+tan(theta2)cos(theta1)
    and got
    mg/sin(theta1)+tan(theta2)*cos(theta1)
     
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