A cd has a mass of 17 g and a radius of 6.0 cm. When inserted into a player, the cd starts from rest and accelerates to an angular velocity of 21 rad / s in 0.8s, assuming the cd is a uniform solid disk. a) Determine the sum of the net torque action on it.

the net toque is τ=8.03 * 10⁻⁴ N*m

Explanation:

Assuming the disk has constant density ρ, the moment of inertia I of is

I = ∫r² dm

since m = ρ*V = ρπR² h, then dm = 2ρπh r dr

thus

I = ∫r²dm = ∫r²2ρπh r dr = 2ρπh ∫r³ dr = 2ρπh (R⁴/4 - 0⁴/4) = ρπhR⁴ / 2 = mR²/2

replacing values

I = mR²/2 = 0.017 kg * (0.06 m) ²/2 = 3.06 * 10⁻⁵ kg*m²

from Newton's second law applied to rotational motion

τ = Iα, where τ=net torque and α = angular acceleration

since the angular velocity ω is related with the angular acceleration through

ω = ωo + α*t → α = (ω-ωo) / t = (21 rad/s-0) / 0.8 s = 26.25 rad/s²

therefore

τ = Iα = 3.06 * 10⁻⁵ kg*m²*26.25 rad/s² = 8.03 * 10⁻⁴ N*m