Question

A uniform disc of diameter 300 mm and of mass 5 kg is mounted on one end of an arm of length 600 mm. The other end of the arm is free to rotate in a universal bearing. If the disc rotates about the arm with a speed of 300 r.p.m. clockwise, looking from the front, with what speed will it precess about the vertical axis?

(a) 14.7 rad/s

(b) 15.7 rad/s

(c) 16.7 rad/s

(d) 17.7 rad/s

I got this question in an international level competition.

The query is from Effect of Gyroscopic Couple on an Aeroplane topic in section Gyroscopic Couple and Precessional Motion of Machine Dynamics

Answer 1

Correct answer is (c) 16.7 rad/s

The explanation: Given: d = 300 mm or r = 150 mm = 0.15 m ; m = 5 kg ; l = 600 mm = 0.6 m ;

N = 300 r.p.m. or ω = 2π × 300/60 = 31.42 rad/s

We know that the mass moment of inertia of the disc, about an axis through its centre of gravity and perpendicular to the plane of disc,

I = m.r^2/2 = 5(0.15)^2/2 = 0.056 kg-m^2

and couple due to mass of disc, C = m.g.l = 5 × 9.81 × 0.6 = 29.43 N-m

Let ωP = Speed of precession

We know that couple (C),

29.43 = I.ω.ωP = 0.056 × 31.42 × ωP = 1.76 ωP

ωP = 29.43/1.76 = 16.7 rad/s