Question

The acceleration $a$ of the plank $P$ required to keep the center $C$ of a cylinder in fixed position during the motion is (no slipping takes place between cylinder and plank) :

• A. $\frac{g}{2}sin\theta$
• B. $2gsin\theta$
• C. $gsin\theta$
• D. $\sqrt{2g}sin\theta$

Answer 1

The correct option is B $2gsin\theta$

The cylinder is rolling without slipping, which means that the surface is moving backwards with the same acceleration of the cylinder.

Let the angular acceleration be $\alpha$,

$\therefore$ torque equation:

$m\times g{}^{\prime }\times r=I\times \alpha$

$I$ is the moment of inertia of the cylinder about the axis passing through it's center.

$\Rightarrow mgsin\theta r=\frac{m{r}^2}{2}\times \alpha$

$\Rightarrow \alpha =2gsin\theta /r$

For, pure rolling $a=\alpha \times r$

$\therefore a=2gsin\theta$