Finding mass of sphere with density

In summary, the conversation discusses determining the mass of a solid ball with a given density function. The method involves converting to spherical coordinates and taking the integral of the density function. The boundaries for the integral are given as 0 to pi for phi, 0 to 2pi for theta, and 0 to a for p. The resulting triple integral is (2a-p)p^2sin(phi)dphi(dtheta)(dphi).
  • #1
burton4
1
0
http://tgnot.com/triple_integral_sphere.jpg

A solid ball of radius a has density given by (2a-p) where p = sqrt(x^2 + y^2 + z^2). Determine its mass.

a) I think it's properly setup by converting to spherical, still iffy if (2a-p)=(2p-p)=p

b) What did I just find and how do I then get mass?Thanks-a-centillion if you reply, this isn't homework, but is part of a practice midterm.
 
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  • #2
(2a-p)=p is only true where a=p. That's on the outside of your sphere. On the inside it's just (2a-p). Integrate that.
 
  • #3
If you take the integral of (2a-p) it should work.
Remember the mass is the integral of the density so the boundaries are p^2sin(phi)dp(dtheta)(dphi)
The boundary conditions are:
phi: 0... pi
theta: 0...2pi
p: 0...a

So, triple integral(2a-p)p^2sinphi(dpdthetadphi)

Sorry, I don't know how to make symbols on here yet but hope this helps!
 

1. How do you find the mass of a sphere with a given density?

To find the mass of a sphere with a given density, you can use the formula: mass = density x volume. First, calculate the volume of the sphere using the formula: volume = (4/3) x π x (radius)^3. Then, multiply the volume by the given density to get the mass of the sphere.

2. What is the formula for finding the volume of a sphere?

The formula for finding the volume of a sphere is: volume = (4/3) x π x (radius)^3. This formula is derived from the formula for finding the volume of a cone, where the cone is divided into small slices and the volume of each slice is calculated using the formula for a cylinder. As the number of slices approaches infinity, the formula converges to the volume of a sphere.

3. Can you find the mass of a sphere without knowing its density?

No, you cannot find the mass of a sphere without knowing its density. Density is a crucial factor in calculating the mass of an object. Without knowing the density, you cannot use the formula mass = density x volume to find the mass of the sphere.

4. How does the density of a sphere affect its mass?

The density of a sphere directly affects its mass. The higher the density, the greater the mass of the sphere will be. This is because density is defined as the amount of mass per unit volume. Therefore, a sphere with a higher density will have more mass packed into a given volume compared to a sphere with a lower density.

5. Can the mass of a sphere change if its density changes?

Yes, the mass of a sphere can change if its density changes. This is because density and mass are directly proportional to each other. If the density of a sphere increases, its mass will also increase. Similarly, if the density decreases, the mass will also decrease. This relationship is defined by the formula mass = density x volume.

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