Mass of sphere using areal density which varies linearly w/ dist. from x,y-plane

polebo

New member
Joined
May 17, 2018
Messages
2
Hello. Would really appreciate some help with this problem; the full solution or guidance.

Problem
The sphere x2+y2+z2 = a2 has a areal density which varies linearly with the distance from the xy-plane. Areal denisty σ = k*abs(z)/a, where k is a constant [mass/area]. Calculate the total mass.

My tries
I know I should use sphereical coordinates:
x = a * sin(φ) * cos(θ)
y = a * sin(φ) * sin(θ)
z = a * cos(φ)

I know that I should get a double integral that has dφdθ at the end. The thing is, I don't know what my limits will be, or what I will integrate. I know I can't integrate the areal density right of the bat.
 
Hello. Would really appreciate some help with this problem; the full solution or guidance.

Problem
The sphere x2+y2+z2 = a2 has a areal density which varies linearly with the distance from the xy-plane. Areal denisty σ = k*abs(z)/a, where k is a constant [mass/area]. Calculate the total mass.

My tries
I know I should use sphereical coordinates:
x = a * sin(φ) * cos(θ)
y = a * sin(φ) * sin(θ)
z = a * cos(φ)

I know that I should get a double integral that has dφdθ at the end. The thing is, I don't know what my limits will be, or what I will integrate. I know I can't integrate the areal density right of the bat.
You do not need to use spherical co-ordinates.

Consider a disk at a distance Z from the origin (center) with a thickness dz (made by slicing through the sphere parallel to the x-y plane).

The radius of the circle = Sqrt(a^2 - z^2)

Consider the upper half only (z positive)

Mass of disk = pi * (a^2 - z^2) * k * z/a * dz

Now integrate....
 
You do not need to use spherical co-ordinates.

Consider a disk at a distance Z from the origin (center) with a thickness dz (made by slicing through the sphere parallel to the x-y plane).

The radius of the circle = Sqrt(a^2 - z^2)

Consider the upper half only (z positive)

Mass of disk = pi * (a^2 - z^2) * k * z/a * dz

Now integrate....

Thank you for your answer!

Yeah, I've tried that. If I integrate with limits 0 and a I get: (pi*k*a^3)/2 and that is the wrong answer.

Did you by any chance miss that it is with areal denisty (with unit [mass/area = kg/m^2]) and not regular density (with unit [mass/volume = kg/m^3])? Or am I doing something wrong?

I think I need something in my integral besides σ that takes the area into account.
 
Top