What is the precession rate?

The angular velocity of precession is given by:

ω_p = mgr/Iω_s

where ω_p is the angular velocity of precession

I is the moment of intertia

ω_s is the angular velocity of the spin

- so Iω_s is the angular momentum, L

m is the mass

g is the acceleration due to gravity

- so mg is the force due to gravity

r is the lever arm (sometimes l - lowercase L - but I've written it r to distinguish it from I - uppecase i - which is the inertia) - the distance of the centre of mass from the support, perpendicular to the applied force

- so mgr is the torque, τ (Greek lowercase tau)

The equation can also therefore be written as:

ω_p = τ/L

For a solid disc, I is given by:

I = mR^2/2 (here R is the radius of the disc, 50 cm, written in capitals to distinguish it from the lever arm)

Substituting this into the equation for the angular velocity of the precession, we get:

ω_p = mgr/(mR^2/2)ω_s

Which handily means that the mass - which we don't know - cancels out:

ω_p = 2gr/R^2 ω_s

As the gyroscoe is mounted at the centre of an 11 cm long axle, r = 11/2 =5.5 cm = 0.055 m (assuming, in the absence of other information, that the axle is perpendicular to the force, i.e. it is horizontal for a force due to gravity)

We are given ω_s = 1000* 2π/60 rad/s = 105 rad/s

We are given R = 50 cm = 0.5 m

We know that g = 9.8 m/s^2 (unless your gyroscope is on the Moon, or some other body - but we'll assume that if we weren't on the surface of the Earth the question would have mentioned it!)

Putting all these numbers in:

ω_p = 2 * 9.8 * 0.055 / (0.5^2 * 105)

ω_p = 0.041 rad/s

Which is (thankfully) the answer given.