For example, let’s consider a thin infinite sheet whose density depends only on the radial distance r from the center of the plate, σ(r) = ae−r/b . Find the total mass of the infinite r sheet. Hint:
For example, let’s consider a thin infinite sheet whose density depends only on the radial
distance r from the center of the plate, σ(r) = ae−r/b . Find the total mass of the infinite r
sheet.
Hint: you should use polar coordinates to solve this problem, where dA = rdrdθ, where integrating over θ should just result in 2π
The last two parts deal with volume density. Consider a solid spherical object of radius R.
- (e) Assume the sphere has a uniform volume density of ρ0. Calculate the mass of the sphere.
- (f) Now assume that the sphere has a non-uniform density which varies with position, such that ρ(r, θ, φ) = ar sin θ. Calculate the total mass of the sphere.
- Hint: in spherical coordinates, dV = r2 sin(θ)drdθdφ, where θ is given as the angle between the z-axis and ⃗r. This convention of naming θ and φ is sometimes opposite from what they use in math classes.
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