To find volume of a cylindrical shell with holes around it

Click For Summary

Discussion Overview

The discussion revolves around calculating the volume and surface area of a cylindrical shell that has multiple spherical holes throughout its structure. Participants explore the mathematical formulation required for this generalized case, focusing on the geometry of the shell and the holes.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant describes the problem of finding the volume and surface area of a cylindrical shell with numerous spherical holes, emphasizing that it is not a homework problem.
  • Another participant asks for the formula for the volume of a cylinder, indicating a need for foundational understanding.
  • A participant provides the formula for the volume of a cylinder but clarifies that they need the volume of the shell, not the cavity inside.
  • There is a discussion about the difference between the volume of the shell and the cylindrical cavity, with a request for clarification on the thickness of the rim and the size of the holes.
  • One participant suggests a model where the cylinder has an inner diameter R and outer diameter R+dR, with holes penetrating the thickness of the shell, and seeks to determine the volume of this modified shell.
  • Another participant questions the size of the holes relative to the pipe and whether they can be assumed to be flat if small.
  • A participant asserts that the holes cannot be assumed to be flat, indicating a need for more complex modeling.
  • There is a suggestion to model the holes as cylindrical sections, with a proposal to orient the pipe and define the geometry mathematically for volume calculation.
  • A mathematical expression for the volume of material removed from the shell due to the holes is presented, along with a suggestion to use software for computation if specific dimensions are provided.

Areas of Agreement / Disagreement

Participants express differing views on the assumptions regarding the size and shape of the holes, and there is no consensus on the best approach to calculate the volume of the cylindrical shell with holes.

Contextual Notes

Participants have not reached a definitive conclusion on the volume calculation, and there are unresolved questions regarding the assumptions about the holes and their impact on the overall volume of the shell.

abhi486
Messages
4
Reaction score
0
ITS NOT A HOMEWORK PROBLEM
PROBLEM::- THINK OF A CYLINDRICAL SHELL WITH NUMEROUS SPHERICAL HOLES ALL AROUND IT.
(FOR EG. http://sell.lulusoso.com/upload/20120317/Underground_Water_Pipe.jpg)
HOW TO FIND VOLUME & SURFACE AREA OF SUCH A CYLINDRICAL SHELL.
(A GENERALIZED CASE FOR N HOLES AROUND THE SHELL.)
 
Last edited by a moderator:
Physics news on Phys.org
What's the formula for the volume of a cylinder?
 
(Pi).r^2. h

& i need the volume of the shell, not the cylindrical cavity.
 
OK, I'm not totally sure how they differ. Do you mean the amount of material required to make one of the things you describe.

Alternatively what is the thickness of the "rim" of the cylinder, and how big are the holes wrt to the cylinder?
 
... there's a cylinder with inner dia R and outer dia R+dR... it has holes all around penetrating the whole thickness of the shell... assume the rim to be hollow and radius of holes to be "r" ...
Now, as we knw, the volume of a normal cylindrical shell is (pi)(R+dr-R)^2.(h).. Now what i want to ask is that --what would be the volume of the cylindrical shell with holes all around... (i have attached a picture of water-pipe, u can see that for reference.)
 
How big are the holes compared to the pipe? If they're small, can we make an assumption that they are flat?
 
no, we can't ...
 
Can you assume the holes were drilled with a bit so the sides of the holes are portions of cylinders and the projection of the holes perpendicular to their axes are circles?
 
Let's assume the pipe has inner radius ##a## and outer radius ##b## and orient the pipe so the surfaces are ##x^2+z^2 = a^2## and ##x^2+z^2 = b^2##. Now we drill a vertical hole of radius ##c## centered on the ##z## axis through the upper side of the cylinder. The volume of material removed is$$
\int_0^{2\pi}\int_0^c\int_{\sqrt{a^2-r^2\cos^2\theta}}^{\sqrt{b^2-r^2\cos^2\theta}}r\, dzdrd\theta$$All you have to do is work that out and multiply by the number of holes. :-p

[Edit, added] If you have numbers for ##a,b,c## Maple will crank out an answer.
 
Last edited:

Similar threads

Undergrad Finding this volume without cylindrical shells
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Question about Integrals to Determine Volume vs Surface Area
  • · Replies 5 ·
Replies
5
Views
4K
Shell in a cylindrical capacitor
  • · Replies 1 ·
Replies
1
Views
2K
How Does Gauss' Law Apply to an Insulated Cylindrical Shell and Rod System?
  • · Replies 5 ·
Replies
5
Views
2K
Volume of a sphere with a cylindrical hole
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Cylindrical shell and cross-section volume concept question
  • · Replies 1 ·
Replies
1
Views
4K
A neutral conducting cylindrical shell
  • · Replies 2 ·
Replies
2
Views
3K
Electric field of spherical shell
  • · Replies 2 ·
Replies
2
Views
2K
Volume of revolution, region bounded by two functions
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Volume created by the revolution of a polar function around the initial line
  • · Replies 1 ·
Replies
1
Views
4K