The figure shows a system consisting of (i) a ring of outer radius rolling clockwise without slipping on a horizontal surface with angular speed and (ii) an inner disc of radius rotating anti-clockwise with angular speed . The ring and disc are separat

Question

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 $ω$ and (ii) an inner disc of radius $2 R$ rotating anti-clockwise with angular speed $ω / 2$ . 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 $30 °$ with the horizontal. Then with respect to the horizontal surface, [2012]

Smart Achievers · Accepted Answer

(1, 2)

Velocity at centre $' O'$ $∴$ $ω / 2$

${\vec{V}}_{P} = 3 R ω \hat{i} - \frac{R ω}{2} \sin 30 ° \hat{i} + \frac{R ω}{2} \cos 30 ° \hat{k}$

$∴$ ${\vec{V}}_{P} = [3 R_{ω} \hat{i} - \frac{R_{ω}}{4} \hat{i}] + \frac{\sqrt{3} R_{ω}}{4} \hat{k}$

$or, {\vec{V}}_{P} = \frac{11}{4} R_{ω} \hat{i} + \frac{\sqrt{3}}{4} R_{ω} \hat{k}$

Solution

1 the point O has linear velocity $3 R ω \hat{i}$

2 the point P has linear velocity $\frac{11}{4} R ω \hat{i} + \frac{\sqrt{3}}{4} R ω \hat{k}$

3 the point P has linear velocity $\frac{13}{4} R ω \hat{i} - \frac{\sqrt{3}}{4} R ω \hat{k}$

4 the point P has linear velocity $(3 - \frac{\sqrt{3}}{4}) R ω \hat{i} + \frac{1}{4} R ω \hat{k}$

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Solution

1 the point O has linear velocity 3Rωi^

2 the point P has linear velocity 114Rωi^+34Rωk^

3 the point P has linear velocity 134Rωi^-34Rωk^

4 the point P has linear velocity (3-34)Rωi^+14Rωk^

Ans. (1, 2) Velocity at centre 'O' ∴ v→O=3Rωi^ V→P=3Rωi^-Rω2sin30°i^+Rω2cos30°k^ ∴ V→P=[3Rωi^-Rω4i^]+3Rω4k^ or, V→P=114Rωi^+34Rωk^

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1 the point O has linear velocity $3 R ω \hat{i}$

2 the point P has linear velocity $\frac{11}{4} R ω \hat{i} + \frac{\sqrt{3}}{4} R ω \hat{k}$

3 the point P has linear velocity $\frac{13}{4} R ω \hat{i} - \frac{\sqrt{3}}{4} R ω \hat{k}$

4 the point P has linear velocity $(3 - \frac{\sqrt{3}}{4}) R ω \hat{i} + \frac{1}{4} R ω \hat{k}$