Finding Frictional Force and Time to Stop a Spinning Disk

[Destrio]

A uniform disk of radius R and mass M is spinning with angular speed ωi. It is placed on a flat horizontal surface; the coefficient of kinetic friction between the disk and surface is μk.
a) Find the frictional force on the disk.
b) How long will it take for the disk to come to rest.

I'm confused about the directions of the forces.
I know mg will be down. And there will be no forces in the x direction.

Ff = μN

Will the normal force be up, opposite and equal to the force of gravity, or will the rotation change that?

Since we don't know the direction the disk is spinning can we find the direction of ωi and α?

Can we just choose for the direction of ωi and α to be downwards from the torque of the disk?

Thanks

 

[Destrio]

If I have τ = Iα
Can I assume α is down?

so I can have:
ΣF = τ - f

or would it be:
ΣF = τ + mg - f - N

Iα + mg = f + N
I = ω/R
ωα/R + mg = f + N

 

[Destrio]

another try:

τ = f x R
τ = Iα
τΔt = I(ω - ωi)
fRΔt = (1/2)MR^2(ω - ωi)
fΔt = (1/2)MR(ω - ωi)

Δt = (1/2)MR(ω - ωi) / μkFn
Δt = MR(ω - ωi) / 2μkMg
Δt = R(ω - ωi) / 2μkg

is this correct?

thanks