# How to solve Linear momentum L using vector and scalr properties?

• Philowns
In summary, the conversation was about finding the orbital angular momentum L in terms of m, r, and w. The correct solution involves substituting the given equations and simplifying using the identity for cross product. The final result is that L = 0, meaning there is no angular momentum when the linear and angular velocities are parallel.
Philowns

## Homework Statement

Orbital angular momentum L is given by L = mr x v
Linear and angular velocity are related by the eqn. v=w x r
Solve for L in terms of m, r, and w.

## Homework Equations

I was thinking of using Lagrange's formula. A x (B x C) = B(A*C)-C(A*B)

## The Attempt at a Solution

well I substituted for v first:
(L/m) = r x (w x r)
= w(r*r) - r(r*w)

I'm guessing that could be simplified, but I don't know how.

L = m[wr^2- r(r*w)] is the best I got.

Your attempt at solving for L in terms of m, r, and w is a good start. However, there are a few errors in your solution. Here is a corrected solution:

First, let's substitute the given equation for v into the equation for L:

L = m(r x v)
L = m(r x (w x r))

Next, we can use the identity A x (B x C) = B(A*C)-C(A*B) to simplify the expression in parentheses:

L = m(r x (w x r))
L = m(w(r*r) - r(r*w))

However, there is a mistake in your calculation of the cross product. The correct equation is r x r = 0, since the vectors are parallel. So we can simplify the expression further:

L = m(w(r*r) - r(r*w))
L = m(w(0) - r(0))
L = 0

This may seem like an odd result, but it is actually correct. The orbital angular momentum L is zero because the linear and angular velocities are parallel to each other. In other words, the object is moving in a straight line and has no angular motion.

I hope this helps. Keep up the good work in your studies!

## 1. What is linear momentum?

Linear momentum, denoted as L, is a physical quantity that describes the motion of an object in a straight line. It is defined as the product of an object's mass and velocity, and is a vector quantity, meaning it has both magnitude and direction.

## 2. How can linear momentum be calculated?

Linear momentum can be calculated using the formula L = mv, where m is the mass of the object and v is its velocity. It can also be calculated using vector properties, such as L = p + r, where p is the linear momentum of an object and r is its position vector.

## 3. What are the scalar properties of linear momentum?

The scalar properties of linear momentum include its magnitude, which is equal to the product of an object's mass and speed, and its direction, which is the same as the direction of the object's velocity.

## 4. How can vector properties be used to solve for linear momentum?

Vector properties can be used to solve for linear momentum by breaking down the vector into its components and using trigonometry to determine the magnitude and direction of the momentum. This method is useful when dealing with objects moving in multiple directions.

## 5. What are some real-world applications of linear momentum?

Linear momentum is an important concept in physics and has many real-world applications. It is used in sports to measure the power and speed of athletes, in transportation to calculate the momentum of moving vehicles, and in engineering to design structures that can withstand the force of moving objects.

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