Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing
math_proof
Group Title
center of mass of the upper half of the ball x^2+y^2+z^2<16 (for z>0)
 2 years ago
 2 years ago
math_proof Group Title
center of mass of the upper half of the ball x^2+y^2+z^2<16 (for z>0)
 2 years ago
 2 years ago

This Question is Closed

malevolence19 Group TitleBest ResponseYou've already chosen the best response.1
Assume 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
 2 years ago

malevolence19 Group TitleBest ResponseYou've already chosen the best response.1
So 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.
 2 years ago

math_proof Group TitleBest ResponseYou've already chosen the best response.0
thanks so much man
 2 years ago

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

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

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

math_proof Group TitleBest ResponseYou've already chosen the best response.0
malevolence19 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?
 2 years ago

malevolence19 Group TitleBest ResponseYou've already chosen the best response.1
You 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.
 2 years ago

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

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

math_proof Group TitleBest ResponseYou've already chosen the best response.0
thanks that explains it well
 2 years ago
See more questions >>>
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.