A solid sphere is rolling on a horizontal plane without slipping. If the ratio of the angular momentum about the axis of rotation of the sphere to the total energy of the moving sphere is π:22, then the value of its angular speed will be ______ rad/s.

Angular momentum about the axis through the centre: L=Iω=2/5mR²ω.

Total kinetic energy of a rolling solid sphere: E=1/2mv²(1+2/5)=7/(10)mv², and with v=ω R this is E=7/(10)mR²ω².

L/E=(2/5mR²ω)/(7/(10)mR²ω²)=2/5·(10)/7·1/ω=4/(7ω).

The mass and the radius cancel, leaving the ratio as a pure function of ω.

Setting 4/(7ω)=π/(22) gives ω=(4×22)/(7π)=(88)/(7π).

The 22 and the 7 are the giveaway: taking π=(22)/7, ω=(88)/7·7/(22)=(88)/(22).

ω=4 rad/s.

The two quantities have different dimensions, so their ratio is not dimensionless; the question is really asking you to spot that π≈(22)/7 was built into the numbers.