To calculate the centre of gravity of a spherical cap

In summary, the conversation is about a disagreement with a book answer for solving a question involving an integral with limits from 0 to π/2. The proposed method is to use limits from arcsin(b/a) to π/2 and the formula for the area of a cap is incorrect. The correct expression is found using a different version of the formula from the same source.
  • #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:

IMG_20210330_164738_415.jpg


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:

IMG_20210330_164704_521.jpg


And here is the answer:

IMG_20210330_164731_518.jpg


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:

I1.JPG


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.
 
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  • #2
You can easily see that your answer must be wrong as the limit when ##b \to a## should be ##a##.

Where did you get the expression for the cap area? It does not seem correct as it is non-zero when ##h \to 0##.
 
  • #3
Thanks very much. Yep, the formula for the surface area is wrong. I got it from https://mathworld.wolfram.com/SphericalCap.html. I may have misinterpreted it. I should have noticed that it didn't tend to zero as h tended to zero. Thanks for seeing that. I used instead a different version of the formula from the same page and it checks out in Wolfram Alpha, will derive by hand now.

$$Area_{cap} = 2\,\pi\,r\,h$$

Thanks,
Mitch.
 
  • #4
I see that I did misinterpret the first version of the formula. The a in that version is the radius of the base of the cap , not the radius of the whole sphere.
 
  • #5
So with that, are you now able to find the correct expression?
 
  • #6
Yes, I am. Thanks very much.
 

Related to To calculate the centre of gravity of a spherical cap

1. How do you define the centre of gravity of a spherical cap?

The centre of gravity of a spherical cap is the point at which the entire mass of the cap can be considered to be concentrated, and where the cap would balance if suspended from that point.

2. What is the formula for calculating the centre of gravity of a spherical cap?

The formula for calculating the centre of gravity of a spherical cap is:

x = Rsin(θ)
y = R(1 - cos(θ))

Where x and y are the coordinates of the centre of gravity, R is the radius of the sphere, and θ is the angle between the base of the cap and the centre of the sphere.

3. How does the height of the spherical cap affect the centre of gravity?

The height of the spherical cap does not affect the location of the centre of gravity, as long as the cap is still considered to be a part of the larger sphere. The centre of gravity will always be located at a distance of Rsin(θ) from the base of the cap, regardless of the height.

4. Can the centre of gravity of a spherical cap be located outside of the cap itself?

No, the centre of gravity of a spherical cap will always be located within the cap itself, as long as the cap is still a part of the larger sphere. If the cap is removed from the sphere, the centre of gravity will shift accordingly.

5. Can the centre of gravity of a spherical cap be located at the exact centre of the sphere?

No, the centre of gravity of a spherical cap will never be located at the exact centre of the sphere. It will always be located somewhere along the axis of symmetry of the cap, but not necessarily at the centre of the sphere.

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