Question

A comet orbits the sun (figure). When it is at location 1 it is a distance ${\mathit{d}}_{\mathbf{1}}$from the sun. When the comet is at location 2, it is a distance${\mathit{d}}_{\mathbf{2}}$from the sun, and has magnitude of momentum${\mathit{p}}_{\mathbf{2}}$. (a) when the comet is at location 1, what is the direction of ${\overrightarrow{\mathbf{L}}}_{\mathbf{A}}$? (b) when the comet is at location 1, what is magnitude of ${\overrightarrow{\mathbf{L}}}_{\mathbf{A}}$? (c) When the comet is at location 2, what is the direction of ${\overrightarrow{\mathbf{L}}}_{\mathbf{A}}$? (d) when the comet is at location 2, what is the magnitude of${\overrightarrow{\mathbf{L}}}_{\mathbf{A}}$ ? Later we’ll see that the angular momentum principle tells us that the angular momentum at location 1 must be equal to the angular momentum at location 2.

Short answer

The comet is at location 1, the direction of ${\overrightarrow{L}}_A$is normally outward to the plane of the paper.

The comet is at location 1, the magnitude of${\overrightarrow{L}}_A$ is $=d_1{p}_1\sin \alpha$.

The comet is at location 2, the direction of ${\overrightarrow{L}}_A$is directed out of the plane of the paper.

The comet is at location 2, the magnitude of${\overrightarrow{L}}_A$ is$=d_2{p}_2$

Step-by-step solution

Definition of Angular Momentum.

The rotating inertia of an object or system of objects in motion about an axis that may or may not pass through the object or system is described by angular momentum.

Relation between the linear momentum and the angular momentum is-

(a.) The expression which relates the radius vector, linear momentum of the object, and the angular momentum is,

$\overrightarrow{L}=\overrightarrow{r}\times \overrightarrow{p}$

Here,$\overrightarrow{r}$is the unit vector,$\overrightarrow{p}$is the linear momentum vector, and$\overrightarrow{L}$ is the angular momentum vector.

Use the right-hand rule to find the direction of angular momentum vector.

The figure which shows the directions of linear momentum, radius vectors at locations 1 and 2 is given below.

When comet is at location 1-

When comet is at location 1, by right hand rule, the direction of${\overrightarrow{L}}_A$ is normally outward to the plane of the paper.

At location 1, the magnitude of angular momentum ${\overrightarrow{L}}_A$is

$\begin{array}{rcl}\left|{\overrightarrow{L}}_A\right|& =& |{\overrightarrow{d}}_1\times {\overrightarrow{p}}_1|\\ & =& d_1{p}_1\sin \alpha \\ & & \end{array}$

Therefore, the orbital angular momentum of the comet at location 1 is$d_1{p}_1\sin \alpha$ .

When comet is at location 2-

The direction of angular momentum at the location 2 according to the right-hand rule is directed out of the plane of the paper.

At location 2, the magnitude of angular momentum${\overrightarrow{L}}_A$ is

$\begin{array}{rcl}\left|{\overrightarrow{L}}_2\right|& =& |{\overrightarrow{d}}_2\times {\overrightarrow{p}}_2|\\ & =& d_2{p}_2\sin 90°\\ & =& d_2{p}_2\end{array}$

Therefore, the orbital angular momentum of the comet at location 2 is$d_2{p}_2$.

Hence, the direction and the magnitude of the comet is at location 1 is normally outward to the plane of the paper and$=d_1{p}_1\sin \alpha$ .

The direction and the magnitude of the comet is at location 2 is directed out of the plane of the paper and$=d_2{p}_2$ .