Angular velocity and acceleration

[Nikstykal]

Homework Statement

Small mass sits on a circular revolving table, 200 mm from center. It is given a constant angular acceleration of 2 rad/s. The static coefficient of friction is 0.2. At what angular velocity will the mass start to slip?

Homework Equations

a_r=v²/r
a_r=r''-rθ'²
a_θ=rθ''+2r'θ'
ΣF_i=ma_i
F_f,max=μN

The Attempt at a Solution

I first set up a free body diagram with w_mass pointing downwards, F_f pointing left, normal pointing upwards, with a_r vector pointing to the right.

Since the distance is unchanging, r = 0.2m, r'=0, r''=0. θ''=2rad/s, θ'=2t rad/s + c, θ=t²+ct+d

ΣF_z=ma_z=N-mg, since a_z=0, N=mg
ΣF_r=ma_r=m(r''-rθ'²)=-F_f
Combining: m(r''-rθ'²)=-μmg
Simplifying: -rθ'²=-μg

θ' = ω = √(μg/r)
θ' = √(0.2⋅(9.81m/s²)/0.2m) = 3.13 rad/s

Does this look correct? I never used the given angular acceleration so I feel like I missed something. Thank you.

 

[rpthomps]

That seems right to me. although the acceleration is to the left (which the math seems to identify with already)