JEE Main · 2024 · 08 Apr 2024 · Shift 2 · Rotational Mechanics

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 surface 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 2 ω

m/s, where the value of x is _____.

Answer: 2

Solution

In the rotating frame, the frictionless groove lets only the centrifugal term act radially:

\frac{d v}{d t} = ω^{2} r

, i.e.

v d v = ω^{2} r d r

\int_{0}^{V} v d v = ω^{2} \int_{1}^{3} r d r ⟹ \frac{V ^{2}}{2} = ω^{2} (\frac{9 - 1}{2}) = 4 ω^{2}

V = 22 ω

Comparing with

x 2 ω

x = 2

JEE Main · 2024 · 08 Apr 2024 · Shift 2 · Integer answer

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