A community for students.
Here's the question you clicked on:
 0 viewing
math_proof
 2 years ago
center of mass of the upper half of the ball x^2+y^2+z^2<16 (for z>0)
math_proof
 2 years ago
center of mass of the upper half of the ball x^2+y^2+z^2<16 (for z>0)

This Question is Closed

malevolence19
 2 years ago
Best ResponseYou've already chosen the best response.1Assume it has a density rho. Then: \[M= \iiint \rho dV\] It's easiest in spherical so: \[M=\rho \int\limits_0^{2 \pi} \int\limits_0^{\frac{\pi}{2}} \int\limits_0^4 r^2 \sin(\theta) d r d \theta d \phi\] If rho is constant however then you should expect (V is volume of total sphere): \[M= \rho \frac{V}{2}=\frac{2\rho \pi (4)^3 }{3}=\frac{128 \rho \pi}{3}\] So we get: \[M= \rho \int\limits_0^{2 \pi} d \phi \int\limits_0^{4}r^2 dr \int\limits_0^{\frac{\pi}{2}}\sin(\theta) d \theta=\rho (2 \pi)(\frac{4^3}{3})(1)=\frac{128 \rho \pi}{3}\] Which we were expecting

malevolence19
 2 years ago
Best ResponseYou've already chosen the best response.1So now you need the centroid in each direction. You can see from symmetry that the x and y ones will be zero and you only need find where the z one is. So we need: \[\bar{z}=\frac{\iiint z \rho dV}{M}\] So rho is still constant, pull it out and then you have the integral of z (r cos(theta) in spherical). Integrate that and divide by the mass we found and you're done.

math_proof
 2 years ago
Best ResponseYou've already chosen the best response.0i had trouble setting up the integration part

malevolence19
 2 years ago
Best ResponseYou've already chosen the best response.1No problem. Thats the hardest part of the process :P

math_proof
 2 years ago
Best ResponseYou've already chosen the best response.0how did you get pi/2 for the limit

math_proof
 2 years ago
Best ResponseYou've already chosen the best response.0malevolence19 and how would you integrate z bar since it is z in and there are no integration of z so it would not go away?

malevolence19
 2 years ago
Best ResponseYou've already chosen the best response.1You need to replace z by rcos(theta) which is the coordinate transformation from cartesian to spherical. And I got a pi/2 because the theta angle sweeps from the positive z axis down towards the xy, then towards the z axis; you want it to STOP at the xy plane, this corresponds to theta=pi/2.

malevolence19
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1353370276385:dw

malevolence19
 2 years ago
Best ResponseYou've already chosen the best response.1The theta should read "theta=pi/2" not a range.

math_proof
 2 years ago
Best ResponseYou've already chosen the best response.0thanks that explains it well
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.