# Relationship between angles [closed]

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.

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

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• I figured out the problem after working through BMS's answer. Thanks. – MathMan08 Sep 18 '13 at 21:59

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.