Work Energy Power

JEE Main level, numerical answer, medium

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Question 172 of 183

A circular table is rotating with an angular velocity of ωrad/s\omega \mathrm{rad} / \mathrm{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 m1 \mathrm{~m} on the groove. All the surfaces are smooth. If the radius of the table is 3 m3 \mathrm{~m}, the radial velocity of the ball w.r.t. the table at the time ball leaves the table is x2ωm/sx \sqrt{2} \omega \mathrm{m} / \mathrm{s}, where the value of xx is _____.

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Answer: 2

ac=ω2xvdvdx=ω2x∫0vvdv=∫13ω2xdxv22=ω2[x22]v22=ω22[32−12]v=22ωx=2\begin{aligned} & a_c=\omega^2 x \\ & \frac{v d v}{d x}=\omega^2 x \\ & \int_0^v v d v=\int_1^3 \omega^2 x d x \\ & \frac{v^2}{2}=\omega^2\left[\frac{x^2}{2}\right] \\ & \frac{v^2}{2}=\frac{\omega^2}{2}\left[3^2-1^2\right] \\ & v=2 \sqrt{2} \omega \\ & x=2\end{aligned}

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