How is the magnitude of L calculated here?

[Benjamin_harsh]

Homework Statement

How magnitude of L is calculated here?

Relevant Equations

How magnitude of L is calculated here?

The green dot shows the position of the Earth at the instant the Sun disappears. The distance from the Sun, $d$, is the Earth's orbital distance and the velocity $v$ is the Earth's orbital velocity.

When the Sun disappears the Earth heads off in a straight line at constant velocity as shown by the horizontal dashed line, so after some time $t$ it has moved a distance $x = vt$ as I've marked on the diagram. The question is now how the angular momentum can be conserved.

The answer is that angular momentum is given by the vector equation:

$\mathbf L = \mathbf r \times m\mathbf v$

where $\mathbf r$ is the position vector, $\mathbf v$ is the velocity vector and $\times$ is the cross product. We are going to end up with the vector $\mathbf L$ pointing out of the page and the magnitude of $L$ is given by:

$|\mathbf L| = m\,|\mathbf r|\,|\mathbf v|\,\sin\theta \tag{1}$

but looking at our diagram we see that:

$\sin\theta = \large\frac{d}{|\mathbf r|}$

and if we substitute this into our equation (1) for the angular momentum we get:

$|\mathbf L| = m\,|\mathbf r|\,|\mathbf v|\,\large\frac{d}{|\mathbf r|}\normalsize = m\,|\mathbf v|\,d \tag{2}$

And this equation tells us that the angular momentum is constant i.e. it depends only on the constant velocity $\mathbf v$ and the original orbital distance $d$.

I didn't understand how $|\mathbf L| = m\,|\mathbf r|\,|\mathbf v|\,\sin\theta$ is written? Why $sin$ instead of $cos$ or $tan$?

I only know one formula for angular momentum; $L = mvr$.

 

[haruspex]

I didn't understand how |L|=m|r||v|sinθ is written?

Do you mean, how they get to that equation?
It follows from the nature of the cross product, $|\vec x\times\vec y|=|x| |y||\sin(\theta)|$, where $\theta$ is the angle between the vectors.

 

[Replusz]

Problem Statement: How magnitude of L is calculated here?
Relevant Equations: How magnitude of L is calculated here?

I only know one formula for angular momentum; L=mvrL=mvrL = mvr.

This is only true in special cases.
The general form is L=m r x v where x is the vector cross product, which has

$$|\vec x\times\vec y|=|x| |y||\sin(\theta)|$$

 

[Benjamin_harsh]

Why we have to use vectors here?

Can't we derive Earth's angular momentum without vectors?

 

[Replusz]

Why we have to use vectors here?

Can't we derive Earth's angular momentum without vectors?

You will have to learn about vectors when you get to high school anyways.

You can do it with NOT THINKING about vectors but you are still imolicitly using L= mr x v