Which angle do I need for the torque?

[PhizKid]

Homework Statement

Which angle is relevant to the torque of the cable with the hinge as the axis of rotation? Where is the radius used in the Torque equation?

Homework Equations

Torque = Force * radius

The Attempt at a Solution

I don't know which one to use, they are both angles created by the axis and the cable. And as for the radius, I think it is the red line I drew perpendicular to the cable.

 

[gsmith]

The torque of the cable on the rod will be equal to the tension in the cable times the sine of $\varphi$ times d_h

 

[PhizKid]

sine is opposite over hypotenuse so I get:

Bu we don't know what 'x' is. How do you solve for it in terms of d_v or d_h or L?

 

[haruspex]

Since phi and theta add up to a right angle you can use either one, provided you use it in the right way.
In the diagram, you know d_h and d_v, so you can compute the hypoteneuse. You just need to find the length of the red line. Can you spot some similar triangles?

 

[PhizKid]

I'm not good with finding similar triangles. I can't see any here no matter how I look at it. But is the perpendicular line I drew a bisector of the cable length?

 

[haruspex]

I'm not good with finding similar triangles. I can't see any here no matter how I look at it. But is the perpendicular line I drew a bisector of the cable length?

No.
All you need for two triangles to be similar is two angles that match. Often, one of those angles doesn't merely match in size, it is actually the same angle in the picture.
In the present case, all you have are right angles, the angle the cable makes to the vertical, and simple sums and differences of those.

 

[TSny]

Assuming you know d_v and d_h, use triangle ABC to find angle θ₁ or θ₂. Then you can use a smaller triangle to find the lever arm for the tension.
[EDIT: This is just a suggestion if you are having trouble seeing haruspex's similar triangles. I'm not saying that my suggestion is a better approach.]