- #1

gnits

- 137

- 46

- Homework Statement
- To calculate the centre of gravity of a spherical cap

- Relevant Equations
- comparison of moments

Could I please ask for help as to why I disagree with a book answer on the following question:

Answer given is book is $$\frac{1}{2}(a+b)$$

Here's my proposed method:

Prior to this question there is an example of a similar question:

And here is the answer:

So, to solve my question I propose to solve the same integral but instead of the limits being $$0\,\,to\,\,\frac{\pi}{2}$$ I will use $$arcsin(b/a)\,\,to\,\,\frac{\pi}{2}$$

And for the area of the whole cap I will use the formula $$Area_{cap} = \pi(h^2+a^2)$$ where h is the height of the cap, so in my case h = a - b and so I have $$A_{cap} =\pi((a-b)^2+a^2)$$

Using Wolfram Alpha to solve the integral (for now, to see if I agree with book answer, will derive by hand if it works) have:

So equating moments, this would lead me to:

$$\pi\,a\,w\,(a^2-b^2)\,=\,\pi((a-b)^2+a^2)\,w\,\bar{x}$$

which gives:

$$\bar{x} = \frac{a(a^2+b^2)}{a^2+(a-b)^2}$$

Which is not the book answer.

Thanks for any help,

Mitch.

Answer given is book is $$\frac{1}{2}(a+b)$$

Here's my proposed method:

Prior to this question there is an example of a similar question:

And here is the answer:

So, to solve my question I propose to solve the same integral but instead of the limits being $$0\,\,to\,\,\frac{\pi}{2}$$ I will use $$arcsin(b/a)\,\,to\,\,\frac{\pi}{2}$$

And for the area of the whole cap I will use the formula $$Area_{cap} = \pi(h^2+a^2)$$ where h is the height of the cap, so in my case h = a - b and so I have $$A_{cap} =\pi((a-b)^2+a^2)$$

Using Wolfram Alpha to solve the integral (for now, to see if I agree with book answer, will derive by hand if it works) have:

So equating moments, this would lead me to:

$$\pi\,a\,w\,(a^2-b^2)\,=\,\pi((a-b)^2+a^2)\,w\,\bar{x}$$

which gives:

$$\bar{x} = \frac{a(a^2+b^2)}{a^2+(a-b)^2}$$

Which is not the book answer.

Thanks for any help,

Mitch.