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WBJEEThe figure shows a system consisting of (i) a ring of outer radius...
The figure shows a system consisting of (i) a ring of outer radius \(3 R\) rolling clockwise without slipping on a horizontal surface with angular speed \(\omega\) and (ii) an inner disc of radius \(2 R\) rotating anti-clockwise with angular speed \(\omega / 2\). The ring and disc are separated by frictionless ball bearings. The point \(P\) on the inner disc is at a distance \(R\) from the origin, where \(O P\) makes an angle of \(30^{\circ}\) with the horizontal. Then with respect to the horizontal surface,
Solution:
Velocity at centre 'O' \(\quad \therefore \quad \vec{v}_{o}=3 R \omega \hat{i}\)
\(\vec{V}_{P}=3 R \omega \hat{i}-\frac{R \omega}{2} \sin 30^{\circ} \hat{i}+\frac{R \omega}{2} \cos 30^{\circ} \hat{k}\)
\(\therefore \quad \vec{V}_{P}=\left[3 R_{\omega} \hat{i}-\frac{R_{\omega}}{4} \hat{i}\right]+\frac{\sqrt{3} R_{\omega}}{4} \hat{k}\)
or, \(\quad \vec{V}_{P}=\frac{11}{4} R_{\omega} \hat{i}+\frac{\sqrt{3}}{4} R_{\omega} \hat{k}\)
\(\vec{V}_{P}=3 R \omega \hat{i}-\frac{R \omega}{2} \sin 30^{\circ} \hat{i}+\frac{R \omega}{2} \cos 30^{\circ} \hat{k}\)
\(\therefore \quad \vec{V}_{P}=\left[3 R_{\omega} \hat{i}-\frac{R_{\omega}}{4} \hat{i}\right]+\frac{\sqrt{3} R_{\omega}}{4} \hat{k}\)
or, \(\quad \vec{V}_{P}=\frac{11}{4} R_{\omega} \hat{i}+\frac{\sqrt{3}}{4} R_{\omega} \hat{k}\)
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