Spherical Cap

[Speedle]

How do I calculate the radius ( or diameter) of the curviture of a spherical cap if all I have is the diameter and depth?
If I slice a dish from a sphere, and measure the diameter of the cut and the depth of the dish, how so I calculate the radius if the entire sphere?

 

[Denis]

When in doubt, use ye olde Google:

http://www.google.ca/search?hl=en&sourc ... q=null&oq=

 

[galactus]

The volume of a spherical cap given the radius, a, of the 'dish' and the depth h: $\displaystyle V=\frac{{\pi}h(3a^{2}+h^{2})}{6}$

But $\displaystyle a=\sqrt{h(2r-h)}$. Sub this in for a and we get: $\displaystyle V=\frac{h^{2}{\pi}(3r-h)}{3}$

Solve this for r and we get $\displaystyle r=\frac{3V+h^{3}{\pi}}{3h^{2}{\pi}}$

There is the radius of the sphere in terms of the volume and depth of the cap.

Whittling a little more and resubbing we find the radius of the sphere in terms of a and h is $\displaystyle \boxed{r=\frac{a^{2}+h^{2}}{2h}}$

I was just playing around and derived this. But the idea is that you can derive these things by googling some known formula and making substitutions.

 

[Speedle]

OK. If I am understanding correctly, with r=88 and h=27, the diameter of the origional sphere would be 313.8. Am I getting this?

 

[Speedle]

I think I mis-stated my question. Let me try again. I am trying to find the focal point of a c-band satalite dish
Dish Diameter= 176 Dish Radius (a) = 88 Dish depth (h)= 27
[(88x88)+(27x27)]/(2x27)=r or
8473/54= 156.9
r= 156.9
Am I getting it?

 

[galactus]

Yes, that is totally different. I wish I would have known that before I derived that spherical cap stuff. A satellite dish is NOT a spherical cap. It is a paraboloid.

If the diameter and depth are known, we can place the vertex at the origin and its axis along the x-axis. An equation for it is $\displaystyle y^{2}=4px$.

Where p is the distance from the center of the dish to the focus.


Since the dpeth is 27 inches and the radius is 88, then we have $\displaystyle 88^{2}=4p(27)$

$\displaystyle p=\frac{1936}{27}\approx 71.7 \;\ inches$

 

[Speedle]

Thank you. That helps alot! Now, for the purpose of aiming the dish, can you tell me how to calculate the radius (or diameter( of the origional entire sphere)?

 

[Deleted member 4993]

Thank you. That helps alot! Now, for the purpose of aiming the dish, can you tell me how to calculate the radius (or diameter( of the origional entire sphere)?

Why is sphere coming back to the problem? - it is a paraboloid!

General equation for radius of curvature is:

$\displaystyle R(x) = \frac{\left [1 + (y')^2\right ]^{\frac{3}{2}}}{y"(x)}$

Galactus has derived the eqution y(x) for you.

 

[Speedle]

I keep going back to a sphere because this dish has a constant curve. (Just as if it was sliced off of a ball) By knowing where the center point would, I can space 2 lnb's so that they can each see different satalites without moving the dish.

 

[galactus]

It may look like a part of a sphere, but it is a paraboloid. All satellite dishes are designed that way. I am done.