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. 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.
Figure

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.
$$\begin{gathered} \therefore m{\omega }^{2}R\cos \theta =ma \\ \Rightarrow a={\omega }^{2}R\cos \theta \end{gathered}$$
Let the time taken by the particle to reach the point B be t.
$$\begin{gathered} \mathrm{On}\mathrm{using}\mathrm{equation}\mathrm{of}\mathrm{motion},\mathrm{we}\mathrm{get}: \\ L=ut\mathit{+}\frac{\mathit{1}}{\mathit{2}}a{t}^2 \\ \Rightarrow L=\frac{1}{2}{\omega }^{2}R\cos \theta t^2 \\ \Rightarrow t^{\mathit{2}}=\frac{2L}{{\omega }^{2}R\cos \theta } \\ \Rightarrow t=\sqrt{\frac{2L}{{\omega }^{2}R\cos \theta }} \end{gathered}$$