Calculate the moment of inertia for each scenario: (a) An 80.0 kg skater is approximated as a cylinder with a 0.140 m radius. (b) The skater extends both her arms, each of which is approximated as a 4.00 kg rod with length 0.850 m rotated about its end. (c) Calculate the angular velocity of the skater during scenario (b) if her angular velocity during scenario (a) is 6.75 rad/s.

a) the moment of inertia is 0.784 Kg*m²

b) the moment of inertia is with arms extended is 1.187 Kg*m²

c) the angular velocity in scenario (b) is 4.45 rad/s

Explanation:

The moment of inertia is calculated as

I = ∫ r² dm

since

I = Ix + Iy

and since the cylinder rotates around the y-axis then Iy=0 and

I = Ix = ∫ x² dm

if we assume the cylinder has constant density then

m = ρ * V = ρ * π R²*L = ρ * π x²*L

therefore

dm = 2ρπL * x dx

and

I = ∫ x² dm = ∫ x² 2ρπL * x dx = 2ρπL∫ x³ dx = 2ρπL (R⁴/4 - 0⁴/4) = ρπL R⁴ / 2 = mR² / 2

therefore

I skater = mR² / 2 = 80 Kg * (0.140m) ²/2 = 0.784 Kg*m²

b) since the arms can be seen as a thin rod

m = ρ * V = ρ * π R²*L = ρ * π R²*x

dm = ρ * π R² dx

I1 = ∫ x² dm = ∫ x² * ρ * π R² dx = ρ * π R²*∫ x² dx = ρ * π R² * ((L/2) ³/3 - (-L/2) ³/3)

= ρ * π R²*2*L³/24 = mL²/12

therefore

I skater 2 = I1 + I skater = mL²/12 + mR² / 2 = 8 Kg * (0.85m) ²/12 + (80-8) Kg * (0.140m) ²/2 = 1.187 Kg*m²

c) from angular momentum conservation

I s2 * ω s2 = I s1 * ω s1

thus

ω s2 = (I s1 / I s2) * ω s1 / = (0.784 Kg*m²/1.187 Kg*m²) * 6.75 rad/s = 4.45 rad/s