Moments of Inertia Derivation,

In summary, the conversation was about a problem involving calculating the moment of inertia for different objects using the equation I = ∑miri2. The person needed help with deriving the moment of inertia for the first and fourth objects. They provided their attempt and solutions for other objects. The conversation also mentioned integrating from 0 to R/2 instead of the entire length of the rod in one of the attempts. The final solution involved simplifying the integral and taking out the constants before solving.
  • #1
Ush
97
0

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 = ∑miri2

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
 

Attachments

  • attempt.jpg
    attempt.jpg
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  • question.jpg
    question.jpg
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  • #2
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.
 
  • #3
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 =[
 
  • #4
Ush said:
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?
 
  • #5
another attempt attached
 

Attachments

  • attempt 2.jpg
    attempt 2.jpg
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  • #6
Ush said:
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.
 
  • #7
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?
 
  • #8
Ush said:
could you give me another hint onto how to do the fourth one?
I thought I did:
Doc Al said:
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?
 
  • #9
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
 
  • #10
Ush said:
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.)
 
  • #11
R is constant for a given shell
 
  • #12
Ush said:
R is constant for a given shell
Right! So simplify and complete the integral: I = ∫R2 dm
 
  • #13
if radius is constant. then mass is constant. there is no dr or dm =S
how do i sub dm for something?
 
  • #14
Ush said:
if radius is constant. then mass is constant.
Not sure what you mean. Hint: How do you deal with constants within the integral sign?
 
  • #15
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!
 

What is moments of inertia derivation?

Moments of inertia derivation refers to the process of calculating the moment of inertia for a given object, which is a measure of its resistance to changes in its rotational motion.

What is the formula for moments of inertia?

The formula for moments of inertia is I = ∫r²dm, where I is the moment of inertia, r is the distance of each particle from the axis of rotation, and dm is the mass of each particle.

What is the difference between moment of inertia and mass?

Moment of inertia and mass are two different physical quantities. Mass is a measure of an object's resistance to changes in its linear motion, while moment of inertia is a measure of its resistance to changes in its rotational motion.

What factors affect the moment of inertia of an object?

The moment of inertia of an object is affected by its mass, shape, and distribution of mass. Objects with larger mass, greater distance from the axis of rotation, and more concentrated mass have larger moments of inertia.

What is the significance of moments of inertia in physics?

Moments of inertia play an important role in many physical phenomena, such as rotational motion, angular momentum, and stability of objects. They are also used in engineering and design to determine the strength and performance of rotating objects.

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