Moments of Inertia Derivation,

[Ush]

Homework Statement

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

Homework Equations

I = ∑m_ir_i²

A = area
M = total mass
dm = change in mass
dA = change in area
dr = change in radius

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

 

[Doc Al]

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.

 

[Ush]

I'm not sure how to integrate one so that I'll get 1/12ML²

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 1/12ML²

I still don't understand how to begin the fourth one =[

 

[Doc Al]

I'm not sure how to integrate one so that I'll get 1/12ML²

You've already done the integral (in #2)--the only change is the limits of integration.

I still don't understand how to begin the fourth one =[

I = ∫r² dm. How does r vary as you move around the shell?

 

[Ush]

another attempt attached

 

[Doc Al]

another attempt attached

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.

 

[Ush]

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?

 

[Doc Al]

could you give me another hint onto how to do the fourth one?

I thought I did:

How does r vary as you move around the shell?

I'll rephrase it. What's the distance from the axis of every element of mass dm as you go around the shell?

 

[Ush]

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

 

[Doc Al]

the distance from the axis of every element of mass, dm, is R ?

Exactly. Is R a variable or a constant? (For a given shell.)

 

[Ush]

R is constant for a given shell

 

[Doc Al]

R is constant for a given shell

Right! So simplify and complete the integral: I = ∫R² dm

 

[Ush]

if radius is constant. then mass is constant. there is no dr or dm =S
how do i sub dm for something?

 

[Doc Al]

if radius is constant. then mass is constant.

Not sure what you mean. Hint: How do you deal with constants within the integral sign?

 

[Ush]

if you have a constant then you take it out of the integral.
..oh my

I = ∫R² dm
= R²∫dm
= R² ∑m
= R²M


THANK YOU SO MUCH!