Calculating Polar Moment of Inertia for Automobile Wheel and Tyre

[a3sportback54]

Homework Statement

I'm having a bit of difficulty with this question on SHM:

"An automobile wheel and tyre are suspended in the horizontal plane by a vertical steel rod 5mm in diameter and 2m long, which is bolted to the wheel axis. The wheel is given a small angular deflection, and makes 10 oscillations in 30.2s. Calculate the polar moment of intertia of the wheel and rod. Assume G = 80 GNm^-2.

Homework Equations

I've got the following equations but I don't think I've got the right variables to be able to use them. Is there something I'm missing?

$$\theta=$$$$\frac{TL}{JG}$$

Where $$\theta$$ = torsional deformation, T = torque, L = length, J = polar moment of inertia, G = shear modulus

Also

k = $$\frac{T}{\theta}$$

I'm not sure where the number of oscillations comes into it, although I know period = 0.302s


Any help much appreciated, thanks

 

[Dr.D]

If you know the period, do you know how to find the frequency?

Assuming that you know how to find the frequency (in rad/s), then do you know how that is related to k and J?

You need to be careful here; there are two different J's in this problem. One is the mass moment of inertia of the wheel, the thing you are supposed to find, and the second is tha area moment of inertia of the rod. Pay attention to which is which. It would be a good idea to denote them Ja and Jw or some such just to keep yourself clear on this.

 

[nvn]

a3sportback54: The two relevant equations you listed are correct. Notice you can substitute one of these equations into the other. As the second paragraph of the post by Dr.D points out, you need to list one more relevant equation. I would probably use the nomenclature I = wheel and tyre mass polar moment of inertia, and J = steel rod area polar moment of inertia. Also, period is not 0.302 s; try that one more time. Here are some questions to help you figure out the third relevant equation. What is the relation (equation) between period tn (s) and frequency f (Hz)? What is the relation between frequency f (Hz) and circular frequency omega (rad/s)? What is the relation between omega, k, and I?