Kinetic energy of body rotating at constant angular speed

[Rhi]

Homework Statement

A straight wire has uniform density and total mass M. The wire is bent to form a closed loop, one section of which is a semi-circle of radius a, and the other section the diameter joining the two ends of the semicircle. The body is free to move about the midpoint O of its straight edge. It rotates at constant angular speed about an axis through O in the plane of the body and perpendicular to its straight edge.

Using T=J_ij*ω_i*ω_j/2 or otherwise, show that T=(1/12)Ma^2ω^2((3π+4)/(π+2))

Homework Equations

J_ij=∫ρ(r_kr_kδ_ij-r_ir_j)dV

The Attempt at a Solution

I've worked out that ρ=M/(a(π+2))

I'm struggling with how to do the integrals, as its such an odd shape off to the examples we've done (cubes, etc). I tried to do the semi-circle bit and the diameter bit separately, but then I don't get aything like the answer I'm supposed to. Any pointers would be appreciated! :)

 

[anathema]

Hello,

Thank you for your post. It seems like you are on the right track. When dealing with an odd shape like this, it can be helpful to break it down into smaller, simpler shapes and then add up the contributions from each shape. In this case, you can split the wire into two sections - the semicircle and the diameter.

For the semicircle section, you can use the formula for the moment of inertia of a circular disk, which is J=1/2MR^2. However, in this case, the radius is not constant but varies along the length of the wire. So you will need to use the moment of inertia formula for a solid cylinder, which takes into account the varying radius. This formula is J=MR^2/4, where R is the average radius of the cylinder. You can use this formula to calculate the moment of inertia of the semicircle section.

For the diameter section, you can use the formula for the moment of inertia of a rod about its center of mass, which is J=1/12ML^2, where M is the mass of the rod and L is its length. In this case, the length of the rod is just the diameter of the semicircle, so you can use this formula to calculate the moment of inertia of the diameter section.

Once you have calculated the moment of inertia for each section, you can add them together to get the total moment of inertia for the wire. Then, you can use the formula T=Jω^2/2 to calculate the kinetic energy of the wire.

Hope this helps! Let me know if you have any further questions.