A circular table is rotating with an angular velocity of $$\omega$$ rad/s about its axis (see figure). There is a smooth groove along a radial direction on the table. A steel ball is gently placed at a distance of 1 m on the groove. All the surfaces are smooth. If the radius of the table is 3 m, the radial velocity of the ball w.r.t. the table at the time ball leaves the table is $$x\sqrt{2}\omega$$ m/s, where the value of $$x$$ is _____

The outward centrifugal force on the ball is $$F = m\omega^2r$$

$$a_r = \omega^2 r$$

$$a = v \frac{dv}{dr}$$

$$v \frac{dv}{dr} = \omega^2 r$$

Integrate both sides from the point where the ball is placed (r = 1 m) to the edge of the table (r = 3 m):

$$\int_{0}^{v} v \, dv = \omega^2 \int_{1}^{3} r \, dr$$

$$\frac{v^2}{2} = \omega^2 \left[ \frac{r^2}{2} \right]_1^3$$

$$v^2 = \omega^2 (3^2 - 1^2)$$

$$v^2 = 8\omega^2$$

$$v = \sqrt{8}\omega = 2\sqrt{2}\omega$$
$$x = 2$$