Finding Moment of Interia of a 'Loop' when given Density & Cross sectional area

[anr91]

Homework Statement

I eventually have to solve for maximum angular acceleration of the loop in a magnetic field, and I have gotten everything with the exception of the moment of inertia, so I won't include the emf and B known variables.
Known: a copper wire with a density of [itex]\rho[/itex] = 8960 kg/m³ is formed into a circular loop of radius 0.50 m. Cross sectional area of the wire is 1.00 x 10^-5m².

Homework Equations

I=MR²
(and eventually) [itex]\tau[/itex] = [itex]\alpha[/itex]I

The Attempt at a Solution

I know since mass isn't given, I need to integrate something so I can use the density. However, it's been a really long time since I've integrated, so I'm not very familar with it. I've been unable to find an equation to find the volume of the 'loop,' so I know integration is the only way.

 

[Mike Pemulis]

No integration required. A loop is just a cylinder bent into a circle. Imagine bending the loop back into a "normal" cylinder, and find the volume of that object. (If you don't remember the formula for the volume if a cylinder, it is easy to find).

 

[WatermelonPig]

You can approximate the loop as a circle (line) because the cross section is much smaller than the radius.

 

[Mike Pemulis]

Yes, I forgot to add that part. But first, OP needs to find the mass, which requires finding the volume.

Once that is done you would forget about the finite width and simply use I = MR²

 

[rwisz]

I would suggest using the formula for moment of inertia of a circular hoop, which is I = MR^2. In this case, the mass (M) can be calculated by multiplying the density (\rho) by the volume of the loop. The volume of the loop can be found by multiplying the cross sectional area (A) by the circumference of the loop (2\pi R). Therefore, the equation for moment of inertia in this case would be I = \rho A (2\pi R)^2. This can then be used to solve for the maximum angular acceleration using the equation \tau = \alpha I. I hope this helps in your calculations.