Question

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.

Answer 1

Answer:

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