The hemisphere of radius r is made from a stack of very thin plates such that the density varies with height, r = kz, where k is a constant. Determine its mass and the distance z to the center of mass G.

Accepted Solution

Answer:M = ¼ k π R⁴zG = 8/15 RStep-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 dVEach plate has a radius r and a thickness dz.  So the volume of each plate is:dV = π r² dzSubstituting:dm = ρ π r² dzWe're told that ρ = kz.  Substituting:dm = kz π r² dzNext, 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 = R²r² = R² − z²Substituting:dm = kπ z (R² − z²) dzWe 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 = ∫ dmM = ∫₀ᴿ  kπ z (R² − z²) dzM = kπ ∫₀ᴿ (R² z − z³) dzM = kπ (½ R² z² − ¼ z⁴) |₀ᴿM = kπ (½ R⁴ − ¼ R⁴)M = ¼ k π R⁴Next, to find the center of gravity, we use the weighted average:zG = (∫ z dm) / (∫ dm)zG = (∫ z dm) / MWe already found M, we just have to evaluate the other integral:∫ z dm∫₀ᴿ kπ z² (R² − z²) dzkπ ∫₀ᴿ (R² z² − z⁴) dzkπ (⅓ R² z³ − ⅕ z⁵) |₀ᴿkπ (⅓ R⁵ − ⅕ R⁵)²/₁₅ k π R⁵Plugging in:zG = (²/₁₅ k π R⁵) / (¼ k π R⁴)zG = ⁸/₁₅ R