Question

A cylinder of radius r and mass m rests on two horizontal parallel comers of two platforms. Both the platforms are of the same height. Platform B is suddenly removed. Assume friction between the comer of the platform A and cylinder to be sufficient enough to prevent sliding. Determine angular acceleration of the cylinder immediately after the removal of the platform B

• A. $\frac{g \sin \theta}{3 r}$
• B. $\frac{2 g \sin \theta}{r}$
• C. $\frac{2 g \sin \theta}{3 r}$
• D. $\frac{2 g \cos \theta}{3 r}$

Answer 1

Answer: (C)

Since the cylinder does not slide at the point of contact with the corner of platform $A$, it rotates about fixed axis through the point of contact in subsequent motion. Torque cquation should be used.
Applying the torque equation about the fixed axis through P, we have
$\sum \vec{\tau}_{ P }= I _{ P } \vec{\alpha} \rightarrow mgr \sin \theta= I _{ P } \alpha \, \dots(1)$

Applying the theorem of parallel axes and expression for moment of inertia about centroidal axes, we obtain moment of inertia $I_{p}$ about an axis through the point $P$
$I_{p}=I_{C}+m r^{2} \rightarrow$
$I_{P}=\frac{1}{2} m r^{2}+m r^{2}=\frac{3}{2} m r^{2} \dots (2)$
Substituting $I_{p}$ from eq. $b$. in a., we have
Angular acceleration of the cylinder $\alpha=\frac{2 g \sin \theta}{3r }$