• Support PF! Buy your school textbooks, materials and every day products Here!

Center of Mass and Surface Area (multivariable)

  • Thread starter clairez93
  • Start date
  • #1
114
0

Homework Statement



1. Verify the given moments of inertia and find the center of mass. Assume each lamina has a density of p=1. The problem gives a circle with a radius a.

2. Find the area of the surface of the portion of the sphere x^2 + y^2 + z^2 = 25 inside the cylinder x^2 + y^2 = 9.

Homework Equations





The Attempt at a Solution



1. I already verified the moment, it's the center of mass that's a problem. It gives a picture of a circle with its center at the origin. I would say the center of mass is (0, 0) since its center is at the origin and has a uniform density. Also, I did the math and all and it works out to 0 for x and y as well. The book however says the answer is (a/2, a/2)

2. I wasn't sure about this one. I tried this double integral.

[tex]\int^{2\pi}_{0}\int^{3}_{0}\frac{5}{\sqrt{25 - r^{2}}}r*dr*d\theta[/tex]

(with all the partial derivative and plugging formula and converting to polar steps taken out)
which comes out to 10pi, but the book says the answer is 20pi.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
I really can't argue with your logic on the first one. But on the second one, I think you are only getting the area for the positive z part. There's also a negative z part.
 
  • #3
114
0
I really can't argue with your logic on the first one. But on the second one, I think you are only getting the area for the positive z part. There's also a negative z part.
Does this mean I have to add another double integral with the equation of -5/sqrt 25-r^2 to get the right answer?
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
Does this mean I have to add another double integral with the equation of -5/sqrt 25-r^2 to get the right answer?
Not exactly. You did the z=sqrt(25-r^2) part. If you do the z=(-sqrt(25-r^2)) part (partial derivatives etc.) you'll get the same integral form.
 

Related Threads for: Center of Mass and Surface Area (multivariable)

Replies
2
Views
11K
Replies
18
Views
3K
Replies
2
Views
2K
  • Last Post
Replies
5
Views
1K
Replies
1
Views
946
Replies
10
Views
2K
  • Last Post
Replies
1
Views
1K
Top