What is the derivation of the moment of inertia of a solid sphere?

Click For Summary
SUMMARY

The moment of inertia of a solid sphere is derived by summing the contributions of infinitesimally small disks, each with mass dm. The term (1/2)y² arises from the formula for the moment of inertia of a disk, which is (1/2)MR², where R is the radius of the disk. The derivation process involves integrating these contributions across the entire volume of the sphere. Different approaches to defining mass elements, such as dm = dfg dr versus dI = abc dm, can lead to confusion, but both methods ultimately aim to achieve the same result.

PREREQUISITES
  • Understanding of basic physics concepts, particularly rotational dynamics.
  • Familiarity with calculus, specifically integration techniques.
  • Knowledge of the moment of inertia and its significance in rotational motion.
  • Ability to interpret and manipulate mathematical formulas related to mass and radius.
NEXT STEPS
  • Study the derivation of the moment of inertia for various shapes, focusing on solid and hollow spheres.
  • Learn about the application of integration in physics, particularly in calculating moments of inertia.
  • Explore the differences between various methods of defining mass elements in rotational dynamics.
  • Review the concept of density and its role in deriving moments of inertia for composite bodies.
USEFUL FOR

Students of physics, educators teaching rotational dynamics, and anyone interested in the mathematical foundations of mechanics will benefit from this discussion.

piquante
Messages
1
Reaction score
0
So here I just need help understanding a concept.
I was reading this website's derivation of the moment of inertia of a solid sphere here: http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph3
For the life of me I can't see where the (1/2)y2 term comes from. I would understand y2 because that would be the radius, but the 1/2?
I understand the other steps but I'm very much stuck on this first one.

A related question for anyone who has an opinion: this website starts out these problems by writing dI = abc dm. My physics professor tends to start with dm = dfg dr. Is there a best way to approach these problems? I'm always the most confused at the beginning when I need to define the above relationships. Thanks!
 
Physics news on Phys.org
To find the moment of inertia of the sphere, they had to sum the moment of inertia of many tiny disks of mass dm. The moment of inertia of a disk of mass M is (1/2)MR^2. To see a derivation of the moment of inertia of a disk: http://mikebloxham.com/H7A/I for sphere.pdf
 

Similar threads

Calculating Moment of Inertia for Hollow Cylinder w/ Water
  • · Replies 40 ·
2
Replies
40
Views
6K
What is the "r" in the moment of inertia?
  • · Replies 13 ·
Replies
13
Views
3K
Correct Use of the Parallel Axis Theorem for Moment of Inertia
  • · Replies 2 ·
Replies
2
Views
2K
Moment of Inertia of Solid Sphere - Proof
  • · Replies 3 ·
Replies
3
Views
6K
Moment of Inertia of a hollow sphere with Mass 'M' and radius 'R'.
  • · Replies 16 ·
Replies
16
Views
4K
Find the inertia of a sphere radius R with rotating axis through the center
  • · Replies 17 ·
Replies
17
Views
2K
Moment of Inertia with pulley and two masses on a string (iWTSE.org)
  • · Replies 8 ·
Replies
8
Views
14K
Calculating the moment of inertia of a solid sphere
  • · Replies 3 ·
Replies
3
Views
2K
Moment of inertia (Perpendicular axis theorem)
  • · Replies 1 ·
Replies
1
Views
2K
Deriving the moment of inertia of a sphere
  • · Replies 6 ·
Replies
6
Views
2K