A table with smooth horizontal surface is fixed in a cabin that r… - Vidyadip

Question

A table with smooth horizontal surface is fixed in a cabin that rotates with a uniform angular velocity $\omega$ in a circular path of radius R (In figure). A smooth groove AB of length L(<<R) is made on the surface of the table. The groove makes an angle $\theta$ with the radius OA of the circle in which the cabin rotates. A small particle is kept at the point A in the groove and is released to move along AB. Find the time taken by the particle to reach the point B.

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Answer

The cabin rotates with angular velocity $\omega$ & radius R

$\therefore$ The particle experiences a force $\text{mR}\omega^2$

The component of $\text{mR}\omega^2$ along the groove provides the required force to the particle to move along AB.

$\therefore$ $\text{mR}\omega^2\cos\theta=\text{ma}$

$\Rightarrow\text{a}=\text{R}\omega^2\cos\theta$

length of groove = L

$\text{L}=\text{ut}+\frac{1}{2}\text{at}^2$

$\Rightarrow\text{L}=\frac{1}{2}\text{R}\omega^2\cos\theta\text{ t}^2$

$\Rightarrow\text{t}^2=\frac{2\text{L}}{\text{R}\omega^2\cos\theta}$

$\Rightarrow\text{t}=\sqrt{\frac{2\text{L}}{\text{R}\omega^2\cos\theta}}$

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