How to find centre of gravity for a hemisphere shell?

Click For Summary
SUMMARY

The discussion focuses on calculating the center of gravity (CG) for a hemisphere shell, emphasizing the integration of mass density over the surface. The key formulas for the CG in rectangular coordinates are provided: ycm = ∫ρ(x) f(x)dx and xcm = ∫ρ(x) xf(x)dx. A critical point raised is the discrepancy between the CG results obtained from integration and symmetry arguments, particularly comparing the semicircular wire's CG at (0, 2R/π) versus the hemisphere's CG at (0, 0, R/2).

PREREQUISITES
  • Understanding of mass density integration
  • Familiarity with center of mass (COM) concepts
  • Knowledge of symmetry in physical problems
  • Basic proficiency in calculus, particularly integration techniques
NEXT STEPS
  • Study the integration of mass density for various geometric shapes
  • Learn about the center of mass calculations for three-dimensional objects
  • Explore the implications of symmetry in physics problems
  • Review examples of center of mass calculations in textbooks or online resources
USEFUL FOR

Students preparing for examinations in physics or engineering, educators teaching mechanics, and anyone interested in understanding the principles of center of gravity in geometric shapes.

mick_1
Messages
5
Reaction score
0
How to find centre of gravity for a hemisphere shell??

Can someone show me how to calculate centre of gravity for a hemisphere shell??
 
Physics news on Phys.org
These sound suspiciously like homework problems.
 
It's not homework, I'm preparing for an examination. And I have serious problem solving this assignment. Pleas help me!
 
Integrate mass density over the surface for each axial direction. As far as I can remember for rectangular coordinates its

y_{cm} = \int{\rho(x) f(x)dx}

x_{cm} = \int{\rho(x) xf(x)dx}

but my memory is probably mistaking me. There should be an example of this in your book though.
 
Here's the hint:use the rotation around Oz symmetry to transform your problem into a very simple one:finding the C of M for a semicircle of radius R.You basically need the "z" coordinate of the C of M for the hemisphere,or the "y" coordinate for the C of M for the semicircle.

Daniel.
 


The result obtained from integration and your symmetry argument are NOT the same.
For a semicircular wire, the COM is (0, \frac{2R}{\pi}) while from integration, it is (0,0,R/2). Can anybody explain this?
 


I know this is an old post, but I have been struggling with the same problem for a while now , can anyone explain this?
 

Similar threads

Undergrad Effects of the centre of gravity on aircraft's stability
  • · Replies 4 ·
Replies
4
Views
2K
Electric field & force due to charged insulating hemispherical shells
  • · Replies 12 ·
Replies
12
Views
2K
To find the weight of a hemisphere
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad Centre of Gravity: How Can This Be Possible?
  • · Replies 2 ·
Replies
2
Views
1K
Centre of mass of a solid hemisphere (Feynman way)
  • · Replies 1 ·
Replies
1
Views
3K
High School Differences between Centre of Gravity and Centre of Mass
  • · Replies 7 ·
Replies
7
Views
3K
Finding the Centre of Mass of a Hemisphere
  • · Replies 13 ·
Replies
13
Views
3K
Undergrad How Did Pre-Newtonian Heliocentrists Explain Gravity?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Reduction of the Centre of Gravity
  • · Replies 14 ·
Replies
14
Views
2K
Finding the center of mass of a solid hemisphere
  • · Replies 19 ·
Replies
19
Views
4K