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.

Answer 1

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.

\[\text { On using equation of motion, we get : }\]

\[L = ut + \frac{1}{2}a t^2 \]

\[ \Rightarrow L = \frac{1}{2} \omega^2 R\cos\theta t^2 \]

\[ \Rightarrow t^2 = \frac{2L}{\omega^2 R\cos\theta}\]

\[ \Rightarrow t = \sqrt{\frac{2L}{\omega^2 R\cos\theta}}\]