Tension in a circular wire with force radiating outwards

[hyperddude]

Homework Statement

Let's say there's a uniform circular wire with radius r. There is a uniform force pushing outward from the center in all directions of the wire. This total force is 10N. What is the tension in the wire?

Homework Equations

N/A

The Attempt at a Solution

Let's say the circle is on an xy coordinate axis with its center at the origin. I think the tension is equal to the total force in the y-direction on the top semicircle. The y-component of the force on the bottom semicircle will be equal in magnitude and in the opposite direction.

To find the the force, we integrate. dF = \frac{10}{2\pi r} dL. For the component in the y-direction, dF = \frac{10}{2\pi r} \sin{θ} dL by some simple geometry. dL = rθ, so we now have dF = \frac{10}{2\pi} \sin{θ} dθ. Integrating from θ=0 to θ=\pi, we get that F=\frac{10}{\pi}, which is the tension.

I'm moderately sure my math is correct, but my real question is whether I found the tension! I'm not very sure if this correctly defines the tension in the circle.

 

[haruspex]

dL = r\sin{θ}

You mean dL = r dθ, right?
Your method is right, but you have overlooked one detail. Draw the free body diagram for one half of the loop. What are all the forces?

 

[hyperddude]

Your method is right, but you have overlooked one detail. Draw the free body diagram for one half of the loop. What are all the forces?

Oh, the force I found, \frac{10}{\pi} wouldn't be the tension because the force gets divided evenly into 2. Then the tension would be \frac{5}{\pi}?

 

[haruspex]

Yup.