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Question
A table with smooth horizontal surface is fixed in a cabin that rotates with a uniform angular velocity ω in a circular path of radius R (In the following figure). A smooth groove AB of length L(<<R) is made the surface of the table. The groove makes an angle θ 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 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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Solution
Let the mass of the particle be m.
Radius of the path = R
Angular velocity = ω
Force experienced by the particle = mω2R
The component of force mRω2 along the line AB (making an angle with the radius) provides the necessary force to the particle to move along AB.
\[\therefore m \omega^2 R \cos\theta = ma\]
\[ \Rightarrow a = \omega^2 R\cos\theta\]
Let the time taken by the particle to reach the point B be t.
\[\text { On using equation of motion, we get : }\]
\[L = ut + \frac{1}{2}a t^2 \]
\[ \Rightarrow L = \frac{1}{2} \omega^2 R\cos\theta t^2 \]
\[ \Rightarrow t^2 = \frac{2L}{\omega^2 R\cos\theta}\]
\[ \Rightarrow t = \sqrt{\frac{2L}{\omega^2 R\cos\theta}}\]
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