CAIE A-Level Further Math 3.3.1 Angular speed

Practise angular speed by linking v, r and ω in circular-motion settings where distances and speeds are expressed algebraically.

[Maximum number: 2]

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 $\frac{\nu_{Q}}{x}=\frac{\nu_{P}}{3 a-x}$

M1

$v_{Q}=\frac{2 x \sqrt{a g}}{3 a-x}$

A1

Shown convincingly: angular speeds equal stated.
AG

2

All question bank results loaded

3.3.1—Angular speed question 1