Assume that the turntable deccelerated during time δt before reaching the final angular velocity ( δt is the time interval between the moment when the top disk is dropped and the time that the disks begin to spin at the same angular velocity). what was the average torque, ⟨τ⟩, acting on the bottom disk due to friction with the record?

See below...

Explanation:

Let’s express ⟨α⟩ in terms of ωi , ωf , and Δt. and torque in terms of It , ωi , ωf , and Δt.

STEP 1.  

The rate of change of angular velocity is Angular acceleration.  

The net change in angular velocity is Average angular acceleration divided by the elapsed time.

⟨α⟩ = ω f −ω i/Δt

STEP 2.

Torque is assumed this way

          dω

   τ = I

           dt

.

⟨τ ⟩ = I t (ω f −ω i )/Δt