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Question: Find the relationship between angles $\theta$ and $\phi$ using the equations of equilibrium and solve for $\theta$.

Express your equation for $\theta$ in terms of $\phi$.

Hint: To derive the relationship between angles $\theta$ and $\phi$, you first must write the equations of equilibrium for the pulley. Once you have the equations of equilibrium, you then can isolate the angles and solve the two equations.

I know that $T_D\cos\theta=T_C\cos\phi$. I also know that $T_D\sin\theta=T_C+T_C\sin\phi$.

I tried solving these equations for $\theta$ in terms of $\phi$, I get keep getting stuck.

Diagram

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closed as off-topic by Emilio Pisanty, Jim, Nathaniel, Qmechanic Sep 18 '13 at 17:33

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  • $\begingroup$ I figured out the problem after working through BMS's answer. Thanks. $\endgroup$ – MathMan08 Sep 18 '13 at 21:59
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I think your labeling of forces is making this harder than it ought to be. Be sure all of your forces refer to forces on the pulley, not forces at some other location (such as on the bottom-right anchor, even if they are equal in magnitude... see below).

I assume the rope is massless. If this is true, then the tension force in the rope is uniform throughout. Call it $T_\text{rope}$. Now, on your free-body diagram for the pulley (not the arm and not the bracket), so far you have two forces of magnitude $T_\text{rope}$, one going straight down and another going down-and-right. Be sure to label the angle $\phi$ here.

The only other force on the (massless) pulley is by the supporting arm. Call it $T_\text{arm}$. Now you have three forces on your FBD. Be sure $\theta$ is labeled here.

Use Newton's laws to relate your three forces. If you know what all the forces sum up to, then you can use some pictorial vector math and determine the unknown angle $\theta$ in terms of $\phi$ without doing math.

The short of it is, if you are careful about labeling your forces, then you don't need to know the weight of the hanging mass, or even calculate the tension force.

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  • $\begingroup$ I don't know the weight or mass. I updated with the hint it gives me. $\endgroup$ – MathMan08 Sep 17 '13 at 21:30
  • $\begingroup$ I changed my answer a lot. Give it a go. $\endgroup$ – BMS Sep 17 '13 at 22:19

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