Correct option is (A) \(\frac{5 \omega _0}6\)
Given,
Mass of cylinder = m
Radius of cylinder = R
Angular velocity = ω0
Velocity of center of mass = ω0R
Height h = \(\frac R4\)
Using conservation of angular momentum
Initial angular momentum = Final angular momentum
The angular momentum is the product of the moment of inertia and angular velocity.
\(\mathrm{L = r_{cm} mv_{cm} + I_{cm}\omega}\)
\(\mathrm{\frac 34 mR^2 \omega = (R - \frac R4)mv_0 + \frac 12 mR^2 (\frac{v_0}R)}\)
\(\mathrm{\frac 34 R \omega = \frac34v_0 + \frac 12v_0}\)
\(\mathrm{\omega = \frac 56 \times \frac{v_0}R}\)
\(\omega = \frac{5 \omega _0}6\)
