1. The problem statement, all variables and given/known data I have attached the problem in one file and I have attached my attempt in the second file. I only need help deriving the moment of inertia for the first (1) and fourth (4) objects but I have attached my solutions to the other objects in case it helps jog someones memory onto how to do this =p 2. Relevant equations I = ∑m_{i}r_{i}^{2} A = area M = total mass dm = change in mass dA = change in area dr = change in radius 3. The attempt at a solution attempt is attached -- Thank you for taking the time to read through my problem and helping me solve it, I appreciate your help
Number 1 is exactly the same problem as number 2, just with different limits of integration. In number 4 note that all the mass is at the same distance from the axis.
I'm not sure how to integrate one so that I'll get 1/12ML^{2} I tried doing something similar dm/M = dr/0.5 L because dr starts from the pivot point in the center and max dr will only cover half of the total length. After doing the integration I didn't get 1/12ML^{2} I still don't understand how to begin the fourth one =[
You've already done the integral (in #2)--the only change is the limits of integration. I = ∫r^{2} dm. How does r vary as you move around the shell?
For some reason, you are integrating from 0 to R/2. That's from the center of the rod to one end. But the rod goes from end to end.
oh wow =o I can't believe I missed that. Thanks so much! i understand how to do the first one now =) could you give me another hint onto how to do the fourth one?
I thought I did: I'll rephrase it. What's the distance from the axis of every element of mass dm as you go around the shell?
the distance from the axis of every element of mass, dm, is R ? if R increases, the mass increases because you get a bigger shell dr/R = dm/M ?? =S
if you have a constant then you take it out of the integral. ..oh my I = ∫R^{2} dm = R^{2}∫dm = R^{2} ∑m = R^{2}M THANK YOU SO MUCH!