[Statics - Torques] Determining the tension of a string

[erickbq]

Homework Statement

A 10.0kg sphere is attached to the incline by a horizontal string. Determine the tension in the string.

Homework Equations

(F⃗ net)x=ΣFx=0

(F⃗ net)y=ΣFy=0

Στ=0

τ = (radial disance)(F)

The Attempt at a Solution

1. I began by determining all the relevant forces associated with the sphere, broke them up into components, and place them in a free-body diagram and table.

2. Since I have no information on the friction force and normal force, I then chose a center of rotation (c.o.r.) that would cancel them.

3. This step is the one I'm having trouble with. To determine the torque with respect to the c.o.r., it's my understanding that you can take the shortest, radial distance from the c.o.r. that is perpendicular to the force (which is the arm lever) and multiply it by that force. My issue is I have no idea how to determine those distances. I have a feeling that this question has no need for that information since I'm sure they'll end up canceling out but I just can't get past it.

 

[mfb]

it's my understanding that you can take the shortest, radial distance from the c.o.r. that is perpendicular to the force (which is the arm lever) and multiply it by that force.

That is a possible way to calculate it.
You can express those distances (marked red in the sketch) as function of the sphere radius r and the given angle with trigonometry. The radius will cancel out later as all distances are proportional to it.