Question

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?

Short answer

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

Step-by-step solution

Step 1: Given

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

Determining the concept        

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.

Determining the precession rate of a gyroscope

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.