A ring of mass M and radius R is rotating with angular speed ω about a fixed vertical axis passing through its centre O with two point masses each of mass at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system is and one of the masses is at a distance of from O. At this instant the distance of the other mass from O is:

Engineering

Physics

Angular Momentum and Conservation of Angular Momentum

Question

A ring of mass M and radius R is rotating with angular speed ω about a fixed vertical axis passing through its centre O with two point masses each of mass $\frac{M}{8}$ at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system is $\frac{8}{9} ω$ and one of the masses is at a distance of $\frac{3}{5} R$ from O. At this instant the distance of the other mass from O is:

$\frac{3}{5} R$

$\frac{2}{3} R$

$\frac{4}{5} R$

$\frac{1}{3} R$

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Angular momentum conservation

${MR}^{2} ω = {MR}^{2} (\frac{8}{9} ω) + \frac{M}{8} {(\frac{3}{5} R)}^{2} \frac{8}{9} ω + \frac{M}{8} x^{2} \frac{8}{9} ω$

$R^{2} = \frac{8}{9} R^{2} + \frac{9}{25} \times \frac{1}{9} R^{2} + \frac{1}{9} x^{2}$

$= (\frac{8}{9} + \frac{1}{25}) R^{2} + \frac{1}{9} x^{2} = (\frac{200 + 19}{225}) R^{2} + \frac{1}{9} x^{2}$

$R^{2} {1 - \frac{209}{225}} = \frac{1}{9} x^{2}$

$x = \frac{3 \times 4}{15} R = \frac{4 R}{5}$