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  4. A pencil is placed vertically on a table top with its point end up and its sticky eraser end down. As it falls over from this unstable position, its point of contact with the table remains stationary during its fall, the tangential acceleration of its tip.

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Q. A pencil is placed vertically on a table top with its point end up and its sticky eraser end down. As it falls over from this unstable position, its point of contact with the table remains stationary during its fall, the tangential acceleration of its tip.

System of Particles and Rotational Motion

Solution:

torque on pencil is $\tau=\frac{m g l \sin \theta}{2}=\frac{m l^{2}}{3} \alpha$
$\Rightarrow \alpha =\frac{3 g \sin \theta}{2 l} $
$a_{\text {tip }} =l \alpha=\frac{3}{2} g \sin \theta$
This we can see it exceeds $g$ when $\sin \theta>\frac{2}{3}$