Time for falling down the incline (Rotating Solid Cylinder)

1. The problem statement, all variables and given/known data
A solid cylinder of radius R is spun and then placed on an incline having coefficient of friction μ=tanθ
(θ is the angle of the incline). The solid cylinder continues to spin without falling for time:
(A)Rωo/3gsinθ
(B)Rωo/2gsinθ
(C)Rωo/gsinθ
(D)2Rωo/gsinθ

3. The attempt at a solution
The solution that I could think of was firstly related to what the question said. It says time elapsed until it starts falling down. So I assumed that that means the time for which the cylinder will keep rotating on the place on the incline it was placed when it was first spun.

So, the equations came out to be:
Let ‘m’ be the mass of the cylinder. Then,
mgsinθ+mRα=μmgcosθ
=>gsinθ+Rα=mgcosθ (condition for cylinder to be where it was initially)

But using this equation, there is no way I can find out the time taken for the cylinder to start coming down. And I don’t see any other way to do so either. So need help on this matter. Reply ASAP. Thanks 🙂

Attached Images
File Type: jpg Rotational Motion.jpg (15.3 KB)

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