A circular table is rotating with an angular velocity of ? 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

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Solution

Q.
A circular table is rotating with an angular velocity of ω 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 the ball leaves the table is $x \sqrt{2} ω$ m/s, where the value of x is ______. [2024]

Ans.
(2)

At some instant of time, the particle is at $x$ (location $A$ )

$F = m ω^{2} x$

Now, acceleration, $a = \frac{F}{m} = \frac{m ω^{2} x}{m} \Rightarrow ω^{2} x = v \frac{d v}{d x}$

$ω^{2} \int_{1}^{3} x d x = \int_{0}^{v} v d v$

$ω^{2} {[\frac{x^{2}}{2}]}_{1}^{3} = {[\frac{v^{2}}{2}]}_{0}^{v}$

$\Rightarrow ω^{2} 8 = v^{2} \Rightarrow v = 2 \sqrt{2} ω \Rightarrow x = 2$

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