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A large rectangular box has been rotated with a constant angular velocity $ω_{1}$ about its axis as shown in the figure. Another small box kept inside the bigger box is rotated in the same sense with angular velocity $ω_{2}$ about its axis (which is fixed to floor of bigger box). A particle $P$ has been identified, its angular velocity about $A B$ would be

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$ω_{1}$

$ω_{2}$

$\frac{ω_{2} (r_{2} - r_{1}) + ω_{1} r_{2}}{r_{2}}$

$\frac{ω_{2} (r_{2} - r_{1}) + ω_{1} r_{1}}{r_{2}}$

answer is B.

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Detailed Solution

$\vec{V_{P}},_{C D} = ω {}_{2}{(r_{2} - r_{1})}$ ; ${\vec{V}}_{C D, A B} = ω {}_{1}r_{1}$

${\vec{V}}_{P}_{, A B} = {\vec{V}}_{P}_{, C D} + {\vec{V_{C D}}}_{, A B} = ω_{2} (r_{2} - r) + ω {}_{1}r_{1}$

Angular velocity of $P$ about $AB$

$ω = \frac{v_{P A B}}{r_{2}} = \frac{ω_{2} (r_{2} - r_{1}) + ω_{1} (r_{1})}{r_{2}}$

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