Question

Problem regarding finding the center of mass of a cone of uniform composition: (One moment, posting below.)

Answer 1

"Find the center of mass of the uniform, solid cone of height h, base radius R, and constant density p. (Hint: Integrate over disk-shaped mass elements of thickness dy, as shwon in the figure.)"

I have pretty much no experience with integration and am learning just up to that in Calculus while doing this in Physics, so i'm a little clueless. Here's a diagram:

Answer 2

First thing you need is a proper definition of centre of mass.
Which one do you have?

Answer 3

Will return to this later, have an exam in less than half an hour, but thanks for responding.

Answer 4

assuming center of mass refers to the point at which we can model this thing as a single point ....

thanx to the symmetry and uniformity of it, the center of mass will be someplace along a line running thru the center of it from top to bottom; the remaining part can be similpified to a trianglular crossection

Answer 5

assuming we cut away equal portions, and thereby equal masses from each side, we are left with a plane to play with

Answer 6

when the area of the top and bottom are equal, then you would have determined the center of mass along that last plane

Answer 7

Hi! Sorry, but the triangular cross section misrepresents the initial distribution of mass.
Besides, @Mendicant_Bias rightly says he has to integrate over infinitesimal discs.

Answer 8

hmmm, yeah i see your point. instead of area, we would want to determine equal volumes of the cone, right?

Answer 9

\[\int_0^{c}\pi~[f(x)]^2=\frac12\int_{0}^{h}\pi [f(x)]^2 \]
\[\int_0^{c}[f(x)]^2=\frac12\int_{0}^{h}[f(x)]^2 \]
and solve for c would be my idea