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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. - Physics

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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.

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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Solution 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. Concept: Circular Motion.
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