A sphere of mass m and radius r << R is placed inside a rotating horizontal cylinder of radius R. As the cylinder's angular acceleration, β, gradually increases, find the maximum value of β that enables the sphere to reach from A to the horizontal point B.

Concept Used:

A small sphere of mass m and radius r is placed inside a rotating horizontal cylinder of radius R . The sphere undergoes pure rolling motion inside the cylinder due to friction.

Calculation:

Applying the rolling condition:

⇒ β(R-r) = (g sinθ + β(R-r)) / (1 + 2/5)

⇒ (7/5) β(R-r) - β(R-r) = g sinθ

⇒ θ = sin⁻¹ (2β(R-r) / 5g)

The maximum angular acceleration (β) required for the sphere to reach point B is:

⇒ β = (5g) / (2R)

Correct Option: Option 4