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.