Further Math A2s 3.3.1 Angular speed

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A particle P of mass m is attached to one end of a light inextensible rod of length 3 a. An identical particle Q is attached to the other end of the rod. The rod is smoothly pivoted at a point O on the rod, where O Q=x. The system, of rod and particles, rotates about O in a vertical plane.

At an instant when the rod is vertical, with P above Q, the particle P is moving horizontally with speed u. When the rod has turned through an angle of $60^{\circ}$ from the vertical, the speed of P is $2 \sqrt{a g}$ , and the tensions in the two parts of the rod, OP and OQ, have equal magnitudes.

Show that the speed of Q when the rod has turned through an angle of $60^{\circ}$ from the vertical is

\frac{2 x}{3 a-x} \sqrt{a g} .

Angular speeds of P and Q are equal,
so $\frac{\nu_{Q}}{x}=\frac{\nu_{P}}{3 a-x}$ $v_{Q}=\frac{2 x \sqrt{a g}}{3 a-x}$
Shown convincingly: angular speeds equal stated.
AG

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CAIE A-Level Further Math A2 3.3.1 Angular Speed Questions

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CAIE A-Level Further Math A2 3.3.1 Angular Speed Questions question 1