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.

 

[Nickie]

I appreciate your attempt at solving this problem using mathematical and geometric concepts. However, I would like to point out a few things.

Firstly, the tension in a circular wire with a force radiating outwards is not a simple mathematical calculation. The tension in a wire is a result of the balance between the internal forces within the wire and the external forces acting on it. In this case, the external force is the uniform force pushing outward from the center in all directions.

Secondly, the tension in a wire is a vector quantity, meaning it has both magnitude and direction. In this scenario, the tension will not be equal to the total force in the y-direction on the top semicircle. The tension will vary along the circumference of the wire, and its direction will also vary.

Finally, the approach you have taken to solve this problem assumes that the wire is in equilibrium, which may not always be the case. The tension in the wire will also depend on the material properties of the wire, such as its elasticity and strength.

In conclusion, while your mathematical approach is commendable, it may not accurately determine the tension in a circular wire with force radiating outwards. I would suggest conducting experiments and analyzing the behavior of the wire under different conditions to accurately determine the tension.