Cutting a hollow sphere into 2 piece but not in half

In summary, the conversation discusses finding the center of mass of a uniform sphere that has been cut into two portions. The smaller portion, S, has a height h measured from its flat surface, where h is less than the radius R of the sphere. The conversation also mentions using two tricks, subtraction and direct integration, to find the center of mass of a hollow H that has the same shape and size as S, but with only a thin curved surface. The conversation ends with a disagreement on the expression for the mass of the cap of H, with one person getting h/2 and the other getting h(4R-4) / 3(3R-h).
  • #1
athrun200
277
0

Homework Statement


A uniform sphere of radius R is cut into two portions. The smaller portion S has height h measured from its flat surface, where h<R.

(a) Locate the center of mass of S from its flat surface

(b) Use (a) the locate the centre of mass of a hollow H which has the same shape and same size as S. The hollow has only a thin curved surface and without any flat surface.
(c) Obtain the result of (b) by direct integration.

Homework Equations


This question use 2 tricks to find the center of mass of a hollow.
1. Use substraction to substract a big object by a smaller one and then take limit. (I hope you understand what I mean.)
2. Use direction integration.


The Attempt at a Solution


My answer in part b doesn't agree with part c, can anyone help me to check what happen?
 

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  • #2
I found a different expression for the mass of the cap:
[tex]m=\frac{\rho\pi h^2(3R-h)}{3}[/tex]
 
  • #3
vela said:
I found a different expression for the mass of the cap:
[tex]m=\frac{\rho\pi h^2(3R-h)}{3}[/tex]

In fact I have same expression as you
[itex]\frac{h^{3}}{3}[/itex]-[itex]h^{2}R[/itex]

But when I copy it form messy worksheet to a much neat writting, I try to simplfy the expression in my mind, unforunately, I fail:tongue2:

So my expression for the mass is also correct.
 
Last edited:
  • #4
Did you get it to work out?
 
  • #5
vela said:
Did you get it to work out?

Yes, of course.
I do this question with 2 other classmates, we check a hundred time for our answer. I am sure we get it correct.
 
  • #6
so may i ask for b) and c)

the ans is..h/2 or h(4R-4) / 3(3R-h)

cuz my ans is not same as yours...

btw where the ans for b and c are not the same
 
  • #7
The answers you get for b and c should both be h/2.
 
  • #8
tim94131358 said:
so may i ask for b) and c)

the ans is..h/2 or h(4R-4) / 3(3R-h)

cuz my ans is not same as yours...

btw where the ans for b and c are not the same

what do you get?
 

What is the purpose of cutting a hollow sphere into 2 pieces?

The purpose of cutting a hollow sphere into 2 pieces is to create different shapes or sections that can be used for various purposes. This process may also be necessary for performing experiments or studying the internal structure of the sphere.

How can a hollow sphere be cut into 2 pieces without cutting it in half?

A hollow sphere can be cut into 2 pieces by making a cut along a different plane or axis than the center. This can be achieved by using specialized tools and techniques, such as laser cutting or water jet cutting.

What materials are suitable for cutting a hollow sphere into 2 pieces?

Materials that are suitable for cutting a hollow sphere into 2 pieces include metals, plastics, and ceramics. These materials are strong and durable enough to withstand the cutting process without breaking or shattering.

What are some potential challenges when cutting a hollow sphere into 2 pieces?

One potential challenge when cutting a hollow sphere into 2 pieces is maintaining the structural integrity of the sphere. The cutting process can weaken the structure, making it more susceptible to breaking or collapsing. Another challenge is achieving precision and accuracy in the cut to create two equal pieces.

Are there any safety precautions to consider when cutting a hollow sphere into 2 pieces?

Yes, there are safety precautions to consider when cutting a hollow sphere into 2 pieces. These may include wearing protective gear, using proper tools and techniques, and having a clear understanding of the structural properties of the sphere to prevent any accidents or injuries.

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