A certain gyroscope consists of a uniform disk with a $\mathbf{50}\mathbf{}\mathbf{cm}\mathbf{}$ radius mounted at the centre of an axle that is$\mathbf{11}\mathbf{}\mathbf{cm}\mathbf{}$long and of negligible mass. The axle is horizontal and supported at one end. If the disk is spinning around the axle at$\mathbf{1000}\mathbf{}\mathit{r}\mathit{e}\mathit{v}\mathbf{/}\mathit{m}\mathit{i}\mathit{n}$, what is the precession rate?

$\begin{array}{l}\mathrm{R}=50\text{cm}\\ \mathrm{L}=11\text{cm}\\ \mathrm{r}=\frac{11}{2}=5.5\text{cm}\\ \mathrm{N}=1000\text{rev/min}\end{array}$

Use the formula for precession rate in terms of mass, gravitational acceleration, spin angular frequency, and moment of inertia to find out the precession rate.

Formula are as follow:

$\begin{array}{l}\mathbf{ф}\mathbf{=}\frac{\mathbf{m}\mathbf{\times }\mathbf{g}\mathbf{\times }\mathbf{r}}{\mathbf{I\omega }}\\ \mathbf{\omega }\mathbf{=}\mathbf{2}\mathbf{\times }\mathbf{\pi }\mathbf{\times }\mathbf{f}\end{array}$

where,$\mathbf{f}$ is frequency, $\mathbf{m}$is mass, $\mathbf{g}$is acceleration due to gravity,$\mathbf{r}$ is radius, $\mathbf{I}$is moment of inertia,$\mathbf{\omega }$ is angular frequency and$\mathbf{ф}$ is precession rate.

To find precession rate, first find angular velocity as follows:

$\mathrm{\omega }=2\times \mathrm{\pi }\times \frac{\mathrm{N}}{60}$

Here, $N$ is the frequency in rpm,

$\mathrm{\omega }=2\times \mathrm{\pi }\times \frac{1000}{60}$

So,

$\mathrm{\omega }=104.72\text{rad/s}$

Now, precession rate is given as,

$\begin{array}{l}\mathrm{ф}=\frac{\mathrm{m}\times \mathrm{g}\times \mathrm{r}}{\mathrm{I\omega }}\\ \mathrm{ф}=\frac{2\times 9.81\times 5.5\times 0.01}{{\left(0.5\right)}^2\times 104.72}\\ \mathrm{ф}=0.041\text{rad/s}\end{array}$

Hence,precession rate of a gyroscope is${\text{0.041rad/sec}}^{\text{2}}$.

Therefore, using the concept of precession, the rate of precession and the direction in which the precession occurs can be found.