Find the Angular Acceleration, acceleration and friction force extered

[Northbysouth]

Homework Statement

The circular disk of 595-mm radius has a mass of 16 kg with centroidal radius of gyration = 500 mm and has a concentric circular groove of 220-mm radius cut into it. A steady force T is applied at an angle θ to a cord wrapped around the groove as shown. If T = 41 N, θ = 39°, μs = 0.23, and μk = 0.19, determine the angular acceleration α of the disk, the acceleration a of its mass center G, and the friction force F which the surface exerts on the disk. The angular acceleration α is positive if counterclockwise, negative if clockwise; the acceleration a is positive if to the right, negative if to the left; and the friction force F is positive if to the right, negative if to the left.

I have attached an image of the question

Homework Equations

r_s is the smaller radius of 220 mm

r_L is the larger radius of 595 mm

I = k²
m

The Attempt at a Solution

I started off by drawing a FBD of the circular disk, in which I included the forces of T, mg, N (normal force) and F (friction force)

ƩF_x: ma_x = Tcos(θ) - F

ƩF_y: ma_y = Tsin(θ) - mg + N

ƩM_G: I_Gα = Tr_s - Fr_L

So, now I have three equations but 5 unknowns: a_x, a_y, F, N and α. How do I find the other two equations?

EDIT:

I've just realized that ay = 0. So, could I use F = u*N, where u would be the kinetic of static friction, as the fourth equation?

 

[Northbysouth]

Yeah, I jumped the gun a bit.

It looked more complicated than it was. I've solved it now. Thanks