Chapter 11, Problem 1Q

Figure 11-23 shows three particles of the same mass and the same constant speed moving as indicated by the velocity vectors. Points a, b, c, and d form a square, with point e at the center. Rank the points according to the magnitude of the net angular momentum of the three-particle system when measured about the points, greatest first.

Figure 11-23 Question 1.

To determine

To find:

Ranking the points according to the magnitude of the net angular momentum of the three-particle system.

Answer to Problem 1Q

Solution:

Ranking of magnitude of net angular momentum is

${\mathrm{L}}_{\mathrm{a}}>\left({\mathrm{L}}_{\mathrm{b}}={\mathrm{L}}_{\mathrm{c}}\right)>{\mathrm{L}}_{\mathrm{e}}>{\mathrm{L}}_{\mathrm{d}}$

Explanation of Solution

1) Concept:

We can use the concept of angular momentum to find the net angular momentum about each point given.

2) Formulae:

$\overrightarrow{\mathrm{L}}\mathrm{ }=\mathrm{ }\overrightarrow{\mathrm{r}}\times \mathrm{m}\overrightarrow{\mathrm{v}}$

3) Given:

Three masses and their velocity vectors are given.

4) Calculations:

Angular momentum about any point is,

$\overrightarrow{\mathrm{L}}\mathrm{ }=\mathrm{ }\overrightarrow{\mathrm{r}}\times \mathrm{m}\overrightarrow{\mathrm{v}}$

We can write magnitude as,

$\left|\overrightarrow{\mathrm{L}}\right|=\mathrm{ }\left|\overrightarrow{\mathrm{r}}\right|\left|\mathrm{m}\overrightarrow{\mathrm{v}}\right|\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\theta }$

Where, $\overrightarrow{\mathrm{L}}$ is angular momentum, $\overrightarrow{\mathrm{r}}$ is position vector,$\overrightarrow{\mathrm{v}}$ is velocity vector, $\mathrm{\theta }$ is angle between  $\overrightarrow{\mathrm{r}}\mathrm{ }$ and $\overrightarrow{\mathrm{v}}$. We can write r for $\left|\overrightarrow{\mathrm{r}}\right|\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\theta }$ .

So, we can write $,$

$\mathrm{L}=\mathrm{r}\mathrm{m}\mathrm{v}$.

 The angular momentum is positive for counter clockwise direction and negative for clockwise direction.

Calculation for net angular momentum about point a:

Let $l$ is the side of square.

${\mathrm{L}}_{\mathrm{a}}=\left(\mathrm{m}\mathrm{v}\right)0+\left(\mathrm{m}\mathrm{v}\right)\mathrm{l}+\left(\mathrm{m}\mathrm{v}\right)\mathrm{l}$

${\mathrm{L}}_{\mathrm{a}}=2\mathrm{m}\mathrm{v}\mathrm{l}$

Calculation for net angular momentum about point b:

${\mathrm{L}}_{\mathrm{b}}=\left(\mathrm{m}\mathrm{v}\right)0+\left(\mathrm{m}\mathrm{v}\right)0+\left(\mathrm{m}\mathrm{v}\right)\mathrm{l}$

${\mathrm{L}}_{\mathrm{b}}=\mathrm{m}\mathrm{v}\mathrm{l}$

Calculation for net angular momentum about point c:

${\mathrm{L}}_{\mathrm{c}}=\left(-\mathrm{m}\mathrm{v}\right)\mathrm{l}+\left(\mathrm{m}\mathrm{v}\right)0+\left(\mathrm{m}\mathrm{v}\right)0$

${\mathrm{L}}_{\mathrm{c}}=-\mathrm{m}\mathrm{v}\mathrm{l}$

Calculation for net angular momentum about point d:

${\mathrm{L}}_{\mathrm{d}}=\left(\mathrm{m}\mathrm{v}\right)\mathrm{l}+\left(-\mathrm{m}\mathrm{v}\right)\mathrm{l}+\left(\mathrm{m}\mathrm{v}\right)0$

${\mathrm{L}}_{\mathrm{d}}=0$

Calculation for net angular momentum about point e:

${\mathrm{L}}_{\mathrm{e}}=\frac{\left(-\mathrm{m}\mathrm{v}\right)\mathrm{l}}{2}+\frac{\left(\mathrm{m}\mathrm{v}\right)\mathrm{l}}{2}+\frac{\left(\mathrm{m}\mathrm{v}\right)\mathrm{l}}{2}$

${\mathrm{L}}_{\mathrm{e}}=\frac{\mathrm{m}\mathrm{v}\mathrm{l}}{2}$

Ranking of magnitude of angular momentum is:

${\mathrm{L}}_{\mathrm{a}}>\left({\mathrm{L}}_{\mathrm{b}}={\mathrm{L}}_{\mathrm{c}}\right)>{\mathrm{L}}_{\mathrm{e}}>{\mathrm{L}}_{\mathrm{d}}$

Conclusion:

We can rank magnitude of net angular momentum using angular momentum concept. We can write equation for each point given at corners of the square.