How is the magnitude of L calculated here?

In summary, the conversation discusses the Earth's position and velocity in relation to the Sun disappearing, and how angular momentum is conserved in this scenario. The formula for angular momentum is given as L=mrv where r is the position vector and v is the velocity vector, and the magnitude of L is calculated using the cross product and the angle between the vectors. It is also mentioned that while it is possible to derive angular momentum without using vectors, it is still implicitly used in the formula.
  • #1
Benjamin_harsh
211
5
Homework Statement
How magnitude of L is calculated here?
Relevant Equations
How magnitude of L is calculated here?
246406
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##.
 
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  • #2
Benjamin_harsh said:
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.
 
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  • #3
Benjamin_harsh said:
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
haruspex said:
$$|\vec x\times\vec y|=|x| |y||\sin(\theta)|$$
 
  • #4
Why we have to use vectors here?

Can't we derive Earth's angular momentum without vectors?
 
  • #5
Benjamin_harsh said:
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
 

Related to How is the magnitude of L calculated here?

1. How is the magnitude of L calculated in scientific research?

The magnitude of L, or the magnitude of a vector quantity, is calculated using the Pythagorean theorem. This involves squaring the components of the vector, adding them together, and then taking the square root of the sum. This calculation can also be represented using vector notation, where the magnitude of a vector is denoted by ||v||.

2. What is the significance of calculating the magnitude of L?

Calculating the magnitude of L is important in understanding the strength or size of a vector quantity. It can also provide valuable information in various scientific fields such as physics, engineering, and mathematics.

3. How does the direction of a vector affect the magnitude of L?

The direction of a vector does not affect the magnitude of L, as it only takes into account the length of the vector. However, the direction can be represented by the components of the vector and can be used to calculate the angle or direction of the vector.

4. Can the magnitude of L be negative?

No, the magnitude of L cannot be negative. It is always a positive value, as it represents the length or size of a vector quantity. However, the components of a vector can be negative, which can affect the direction of the vector.

5. How is the magnitude of L different from the magnitude of other vector quantities?

The magnitude of L is specific to a vector quantity, while the magnitude of other vector quantities may represent different characteristics. For example, the magnitude of velocity represents the speed of an object, while the magnitude of acceleration represents the rate of change of velocity.

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