Answer:
M = ¼ k π R⁴
zG = 8/15 R
Step-by-step explanation:
Note: I'm using lower case r as the radius of each plate and upper case R as the radius of the hemisphere.
The mass of each plate is density times volume:
dm = ρ dV
Each plate has a radius r and a thickness dz. So the volume of each plate is:
dV = π r² dz
Substituting:
dm = ρ π r² dz
We're told that ρ = kz. Substituting:
dm = kz π r² dz
Next, we need to write the radius r in terms of the height z. To do that, we need to look at the cross section (see image below).
The height z and the radius r form a right triangle, where the hypotenuse is the radius of the hemisphere R.
Using Pythagorean theorem:
z² + r² = R²
r² = R² − z²
Substituting:
dm = kπ z (R² − z²) dz
We now have the mass of each plate as a function of its height. To find the total mass, we integrate between z=0 and z=R.
M = ∫ dm
M = ∫₀ᴿ kπ z (R² − z²) dz
M = kπ ∫₀ᴿ (R² z − z³) dz
M = kπ (½ R² z² − ¼ z⁴) |₀ᴿ
M = kπ (½ R⁴ − ¼ R⁴)
M = ¼ k π R⁴
Next, to find the center of gravity, we use the weighted average:
zG = (∫ z dm) / (∫ dm)
zG = (∫ z dm) / M
We already found M, we just have to evaluate the other integral:
∫ z dm
∫₀ᴿ kπ z² (R² − z²) dz
kπ ∫₀ᴿ (R² z² − z⁴) dz
kπ (⅓ R² z³ − ⅕ z⁵) |₀ᴿ
kπ (⅓ R⁵ − ⅕ R⁵)
²/₁₅ k π R⁵
Plugging in:
zG = (²/₁₅ k π R⁵) / (¼ k π R⁴)
zG = ⁸/₁₅ R
