Inertia matrix of a homogeneous cylinder

Click For Summary
SUMMARY

The discussion focuses on deriving the inertia matrix of a homogeneous cylinder, specifically the expressions (1/4)mR² + (1/12)ml² and (1/2)mR². Participants clarify that these expressions result from integrating the squared coordinates y'², z'², and x'², which represent the distribution of mass in the cylinder. The integration process involves using the differential mass element ##dm = \rho \, dx'dy'## and applying the appropriate boundaries to compute the total mass from the volume of the cylinder.

PREREQUISITES
  • Understanding of integral calculus
  • Familiarity with the concept of mass distribution
  • Knowledge of the physical properties of homogeneous cylinders
  • Basic understanding of the inertia matrix in physics
NEXT STEPS
  • Learn how to compute integrals involving variable limits and mass distributions
  • Study the derivation of inertia matrices for different geometric shapes
  • Explore the relationship between density, volume, and mass in rigid bodies
  • Investigate the application of the parallel axis theorem in inertia calculations
USEFUL FOR

Students in physics or engineering, particularly those studying dynamics and mechanics, as well as educators looking to explain the derivation of inertia matrices for various shapes.

influx
Messages
162
Reaction score
1

Homework Statement


[/B]
3d45da.png


Homework Equations


N/A

The Attempt at a Solution


What I am confused about is where they got the (1/4)mR^2 + (1/12)ml^2 and (1/2)mR^2 from? I am guessing that these came from the integral of y'^2 + z'^2 and x'^2 +y'^2 but I don't understand how this happened exactly? Could someone point me in the right direction?

Thanks
 
Physics news on Phys.org
Did you try actually computing the integrals? What did youget? Please show your work.
 
Orodruin said:
Did you try actually computing the integrals? What did youget? Please show your work.

What are y'^2 and z'^2? I don't know what to sub in for these (not sure what they represent?) so can't do the integral until I know them.
 
They are coordinates as shown in the figure.
 
Orodruin said:
They are coordinates as shown in the figure.
I understand that but I mean how would one integrate those terms with respect to m? They aren't constants?
 
They depend on where in the body you are.
 
Orodruin said:
They depend on where in the body you are.
Sorry for the late reply. I am still confused. I just want to know how they (mathematically) got from the left hand side to the right. I don't understand how integrating the left hand side yields the right hand side.
 
Use ##dm = \rho \, dx'dy'## and integrate over ##x'## and ##y'## with the appropriate boundaries. The value of ##\rho## in terms of the total mass ##m## can be inferred by computing the volume ##V## of the body and using ##\rho V = m##.
 

Similar threads

Engineering Calculating moment of inertia about z axis
  • · Replies 5 ·
Replies
5
Views
4K
Question about the inertia matrix of a bar
  • · Replies 8 ·
Replies
8
Views
4K
Finding the linear mapping between homogeneous coordinates
  • · Replies 1 ·
Replies
1
Views
2K
Engineering How do I find the transition matrix of this dynamic system?
  • · Replies 18 ·
Replies
18
Views
3K
Engineering Deriving the Cable Equation (neuroscience) from Fundamental Physics Laws
  • · Replies 1 ·
Replies
1
Views
3K
Can somone explain this review solution (moments/inertia)?
  • · Replies 1 ·
Replies
1
Views
2K
Inertia of a beam with added masses
  • · Replies 3 ·
Replies
3
Views
2K
Moment of inertia of a disc using rods as differential elements
  • · Replies 8 ·
Replies
8
Views
2K
Vertical Component Reactions given angular velocity
  • · Replies 1 ·
Replies
1
Views
2K
How to find the diagonal matrix and it's dominant eigenvalue
  • · Replies 18 ·
Replies
18
Views
2K