Precession of a Gyroscope — Question 7

Question 7

Slow spin decay makes precession accelerate

A gyroscope remains in a quasisteady slow-precession regime while $\omega_s(t)=\omega_0 e^{-kt}$, with $k>0$. Its mass, center offset and axial inertia stay fixed. Initially its precession rate is $\Omega_0$. Assume damping varies slowly enough for the approximation over the interval considered.

1. Find accumulated precession angle from time zero.

2. Find the time and accumulated angle when spin has halved, and state a limitation of extrapolating to very late times.

Question 7 - Solution

Slow spin decay makes precession accelerate

(a) Derivation and calculation

With fixed $M,\ell,I_s$, the quasisteady law gives $$\Omega(t)=\frac{Mg\ell}{I_s\omega_0e^{-kt}}=\Omega_0e^{kt}.$$ Integrating, $$\boxed{\phi(t)-\phi(0)=\int_0^t\Omega_0e^{ku}\,du =\frac{\Omega_0}{k}(e^{kt}-1)}.$$

(b) Result and interpretation

Half spin occurs when $e^{-kt}=1/2$: $$\boxed{t_{1/2}=\frac{\ln2}{k}},\qquad \boxed{\Delta\phi(t_{1/2})=\frac{\Omega_0}{k}(2-1)=\frac{\Omega_0}{k}}.$$ The rate ratio grows as $$\frac{\Omega(t)}{\omega_s(t)}=\frac{\Omega_0}{\omega_0}e^{2kt}.$$ At sufficiently late times the assumed slow, spin-dominated regime fails, so the exponential precession prediction cannot be continued indefinitely.

Physical check

At early times the angle reduces to $\Delta\phi\simeq\Omega_0t$, as required by the initial rate.