Calculating Center of Mass in Tank with Muddy Suspension | Homework Help

Click For Summary
SUMMARY

The discussion focuses on calculating the center of mass for a tank shaped as a lower hemisphere with radius R, filled with a muddy suspension characterized by a density function ρ(x,y,z) = e^-h(x,y,z). The height function h(x,y,z) is determined to be z, where z is the vertical coordinate with z = 0 at the tank's bottom. Participants seek clarification on the limits of integration, particularly the phi limits, for solving the problem.

PREREQUISITES
  • Understanding of spherical coordinates and their application in integration.
  • Familiarity with density functions and their role in calculating center of mass.
  • Knowledge of multivariable calculus, specifically triple integrals.
  • Graphical interpretation of geometric shapes, particularly hemispheres.
NEXT STEPS
  • Study the application of spherical coordinates in triple integrals.
  • Learn about density functions and their impact on center of mass calculations.
  • Review examples of calculating center of mass for non-uniform density distributions.
  • Explore graphical methods for visualizing limits of integration in multivariable calculus.
USEFUL FOR

Students in physics or engineering courses, particularly those tackling problems involving center of mass and density in three-dimensional shapes.

vintwc
Messages
24
Reaction score
0

Homework Statement


Let Ω be a tank whose shape is that of the lower hemisphere of radius R. The tank with a muddy suspension whose density ρ is ρ(x,y,z):=e^-h(x,y,z), where h(x,y,z) is the height of (x,y,z) above the lowest point of the tank. Find the center of mass in the tank


Homework Equations





The Attempt at a Solution


First of all, how does one determine the height, h(x,y,z)? I guess it would be R but I am not able to give a reasoning to my guess. I would appreciate if someone could give me a graphical illustration on how to find the limits of integration for this problem as well (ignore this if it will cause too much hassle). Thanks
 
Physics news on Phys.org
If z is the vertical coordinate, and z =0 at the bottom of the tank, then h(x,y,z) = z
 
I got a feeling its z+R. Could anyone let me know what are the phi limits of integration?
 

Similar threads

Finding the Center of Mass in a Tank with Muddy Suspension | Homework Equations
  • · Replies 3 ·
Replies
3
Views
2K
Can Calculating Cylinder Dimensions Verify Tank Chart Accuracy?
  • · Replies 7 ·
Replies
7
Views
2K
Finding the Centre of Mass of a Hemisphere
  • · Replies 13 ·
Replies
13
Views
3K
Engineering Calculating the mass of air in a pressurised air tank
  • · Replies 14 ·
Replies
14
Views
3K
Centre of mass of a hollow pyramid
  • · Replies 13 ·
Replies
13
Views
3K
How Do You Calculate the Volume of a Diagonally Cut Half-Cylinder?
  • · Replies 2 ·
Replies
2
Views
5K
How can we find the center of mass of a solid cone?
  • · Replies 3 ·
Replies
3
Views
1K
High School Mass estimate only through mass rate of change
  • · Replies 4 ·
Replies
4
Views
2K
Finding center of mass of solid
  • · Replies 1 ·
Replies
1
Views
2K
Center of Mass and Mass Calculation
  • · Replies 7 ·
Replies
7
Views
2K