Rotating turntable with a ball

Classical Mechanics Level 4

The turntable shown in figure is rotated with constant ω = 2 \omega = 2 rad/sec. A ball of mass 3 kg is gently placed in a frictionless tunnel at a distance 3 3 3\sqrt3 m from the center. Find the radial velocity of ball (in m/s) when it reaches at the end of the tunnel provided, the length of the tunnel is 14 m.


The answer is 26.

Pankaj Joshi by Pankaj Joshi , 23, India

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1 solution

We know that

F = d p / d t = m d v / d t = m d v / d t d x / d x = m d x / d t d v / d x = m v d v / d x F=dp/dt=mdv/dt= m dv/dt dx/dx= m dx/dt dv/dx= mv dv/dx

And the radial force normal is

F = m V 2 / x = m w 2 x F=m V^2/x=mw^2 x

Equaling these equations we get

v d v / d x = w 2 x vdv/dx=w^2 x so v d v = w 2 x d x vdv= w^2 xdx

Integrating in both sides

v 2 = w 2 ( R 2 − R 0 2 ) v^2=w^2 (R^2 - R0^2) V = w R 2 − R 0 2 = 26 V= w \sqrt{R^2-R0^2}=26 m/s

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