Chapter 7.12, Problem 127RP

To determine

For what ratios of $\left(\frac{l}{r}\right)$ will the motion be direct or retrograde precession?

Answer to Problem 127RP

The motion is in direct precession if $\left(\frac{l}{r}\right)>\sqrt{6}$.

The motion is in retrograde precession if $\left(\frac{l}{r}\right)<\sqrt{6}$.

Explanation of Solution

Write the expression of moment of inertia along x axis.

$\begin{array}{c}{I}_o=I_{xx}\\ =\frac{1}{2}m{r}^2+\frac{1}{12}m{l}^2\\ =\frac{1}{2}m\left(r^2+\frac{l^2}{6}\right)\end{array}$

Here, mass of the cylinder is m, radius of the cylinder is r and length of the cylinder is l.

Write the expression of moment of inertia along z axis.

$I=I_{zz}=m{r}^2$

Obtain the ratio of moment of inertia.

$\begin{array}{l}\frac{I_o}{I}=\frac{\frac{1}{2}m\left(r^2+\frac{l^2}{6}\right)}{m{r}^2}\\ \frac{I_o}{I}=\frac{1}{2}+\frac{1}{12}{\left(\frac{l}{r}\right)}^2\end{array}$

Direct precession:

  $\frac{I_o}{I}>1$

  $\begin{array}{l}\frac{1}{2}+\frac{1}{12}{\left(\frac{l}{r}\right)}^2>1\\ {\left(\frac{l}{r}\right)}^2>6\\ \left(\frac{l}{r}\right)>\sqrt{6}\end{array}$

Thus, the motion is in direct precession if $\left(\frac{l}{r}\right)>\sqrt{6}$.

Retrograde precession:

  $\frac{I_o}{I}<1$

  $\begin{array}{l}\frac{1}{2}+\frac{1}{12}{\left(\frac{l}{r}\right)}^2<1\\ {\left(\frac{l}{r}\right)}^2<6\\ \left(\frac{l}{r}\right)<\sqrt{6}\end{array}$

Thus, the motion is in retrograde precession if $\left(\frac{l}{r}\right)<\sqrt{6}$.