Question

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 dform a square, with point eat 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.

Short answer

Ranking of the magnitude of net angular momentum is

$L_a>(L_b=L_c)>L_e>L_d$

Step-by-step solution

Step 1: Given 

Three masses and their velocity vectors are given.

Determining the concept 

Angular momentum is equal to the moment of linear momentum. The angular momentum is the cross product of the radius vector with the linear momentum.

Use the concept of angular momentum to find the net angular momentum about each point given.

Formulae are as follows:

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

Here,$\overrightarrow{L}$is angular momentum,$\overrightarrow{r}$is the position vector,$\overrightarrow{v}$is velocity, and m is mass.

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

Angular momentum about any point is,

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

We can write magnitude as,

$\left|\overrightarrow{L}\right|=\left|\overrightarrow{r}\right|\left|m\overrightarrow{v}\right|sin\theta$

Here,$\overrightarrow{L}$is angular momentum,$\overrightarrow{r}$is the position vector,$\overrightarrow{v}$is velocity vector,$\theta$is an angle between$\overrightarrow{r}$and$\overrightarrow{v}$. We can write r for$\left|\overrightarrow{r}\right|sin\theta$.

So, it can be written as

$L=rmv$

.

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

Now, calculate net angular momentum about point a:

Let$l$is the side of the square.

$L_a=\left(mv\right)0+\left(mv\right)l+\left(mv\right)l$

$L_a=2mvl$

Now, calculate net angular momentum about point b:

$L_b=\left(mv\right)0+\left(mv\right)0+\left(mv\right)l$

$L_b=mvl$

Now, calculate net angular momentum about point c:

$L_c=(-mv)l+\left(mv\right)0+\left(mv\right)0$

$L_c=-mvl$

Now, calculate net angular momentum about point d :

$L_d=\left(mv\right)l+(-mv)l+\left(mv\right)0$

$L_d=0$

Now, calculate net angular momentum about point e:

$L_e=\frac{(-mv)l}{2}+\frac{\left(mv\right)l}{2}+\frac{\left(mv\right)l}{2}$

$L_e=\frac{mvl}{2}$

Thus, ranking of magnitude of angular momentum is:

$L_a>(L_b=L_c)>L_e>L_d$

Therefore, the magnitude of net angular momentum can be ranked using the angular momentum concept.