Module 13: Define Angular Momentum; Euler’s 2nd Law (The Moment Equation)

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From the course by Georgia Institute of Technology

Engineering Systems in Motion: Dynamics of Particles and Bodies in 2D Motion

162 оценки

Georgia Institute of Technology

Engineering Systems in Motion: Dynamics of Particles and Bodies in 2D Motion

162 оценки

This course is an introduction to the study of bodies in motion as applied to engineering systems and structures. We will study the dynamics of particle motion and bodies in rigid planar (2D) motion. This will consist of both the kinematics and kinetics of motion. Kinematics deals with the geometrical aspects of motion describing position, velocity, and acceleration, all as a function of time. Kinetics is the study of forces acting on these bodies and how it affects their motion. --------------------------- Recommended Background: To be successful in the course you will need to have mastered basic engineering mechanics concepts and to have successfully completed my courses en titled an “Introduction to Engineering Mechanics” and “Applications in Engineering Mechanics.” We will apply many of the engineering fundamentals learned in those classes and you will need those skills before attempting this course. --------------------------- Suggested Readings: While no specific textbook is required, this course is designed to be compatible with any standard engineering dynamics textbook. You will find a book like this useful as a reference and for completing additional practice problems to enhance your learning of the material. --------------------------- The copyright of all content and materials in this course are owned by either the Georgia Tech Research Corporation or Dr. Wayne Whiteman. By participating in the course or using the content or materials, whether in whole or in part, you agree that you may download and use any content and/or material in this course for your own personal, non-commercial use only in a manner consistent with a student of any academic course. Any other use of the content and materials, including use by other academic universities or entities, is prohibited without express written permission of the Georgia Tech Research Corporation. Interested parties may contact Dr. Wayne Whiteman directly for information regarding the procedure to obtain a non-exclusive license.

From the lesson

Work-Energy Principle for Particles/Systems of Particles

In this section students will learn the work-energy principle for particles/systems of particles, impulse and momentum, impact, conservation of momentum and Euler's 2nd Law - Moment of momentum.

Module 9: Work and Kinetic Energy Principle for Particles/System of Particles; Work of a Linear Spring6:21

Module 10: Work Done by Gravity; Work Done by Friction; Solve Work-Energy Problems for Particles/System of Particles8:41

Module 11: Impulse-Momentum Relationship; Define Impact6:30

Module 12: Define Coefficient of Restitution; Solve an Impact Problem11:03

Module 13: Define Angular Momentum; Euler’s 2nd Law (The Moment Equation)7:12

Meet the Instructors

Dr. Wayne Whiteman, PE
Senior Academic Professional
Woodruff School of Mechanical Engineering

[BLANK_AUDIO]

Hi and welcome to Module 13 of Two Dimensional Dynamics.

Today's learning outcomes are to define Angular Momentum, and to

develop Euler's 2nd Law which is also called the Moment Equation.

So, today's lesson is quite a little bit of theory,

but it's important theory that we're going to use throughout the course.

And so I I hope you can stay with me and, and follow each of the steps.

You should be able to. And so, let's look at Euler's 2nd Law.

First of all, let's recall Newton's 2nd Law for an Inertial Reference Frame.

Newton's 2nd Law says that the, some of the forces acting on a particle is equal

to the time derivative of the linear momentum

of that particle, or the momentum of that particle.

Since a particle can only have linear momentum.

And if I take a system

of particles, I have a bunch of particles here.

I can then cross, I can look at the, the system of particles from

some Inertial Frame F, maybe there, the

earth or some fixed frame Inertial Reference Frame.

I can look at it and I can draw a position vector

from the origin of that fixed frame to each of the particles.

I've done that pair, I've crossed that

position vector on both sides of my equation

for Newton's 2nd Law.

And I know from way back in my earlier course introduction

to engineering mechanics, that R cross F is defined as a moment.

So I've got the moment about o is equal to

the time derivative of this stuff on the right hand side.

And this stuff on the right hand side, its the time derivative of

what we define as the angular momentum, and we give it the symbol

capital H.

And so its, its like the moment of momentum.

It's called the moment equation, so we have R cross m v, m v is the momentum,

so we have the moment of momentum or angular momentum.

That's for a fixed referenced frame, let's look for any arbitrary point P.

For an arbitrary point P, the angular momentum about point

P is the angular momentum about the mass center, plus

a position vector from this arbitrary point P to the

mass center, crossed with the linear momentum of the entire body.

And so H sub p again, angular momentum about point

P, H sub C angular momentum about mass center C.

And this is the moment of linear momentum about point P.

So you can actually see that development for

an arbitrary point P in any standard dynamics textbook.

So let's go back now, and look at the system

of particles as referenced from a point fixed in an

Inertial Reference Frame. And this is the, what we had come up with.

So, I have the sum of the moments about o, and

I can use the product rule here to take the derivative.

That's times, the timed derivative of the position vector crossed with m v,

plus the position vector crossed with the timed derivative of m v which is m a.

But you'll note that the position,

the derivative of the position vector Ri, is the same as Vi.

And so, this gives us Vi crossed with Vi, and we know that Vi crossed

with Vi or Ri dot crossed with Vi is then going to be equal to 0.

And so what I'm left with is, Euler's 2nd

law for a point fixed in an Inertial Reference

Frame is the sum of the moments about point

O is equal to the summation of R cross

m a.

Or some of the moments about point O is equal to the time

derivative of the angular momentum of point

O, for the definition of angular momentum.

So that's one result for Euler's 2nd law or The

Moment Equation for a point fixed in an Inertial Reference Frame.

The sum of the moments equals the time

derivative of the angular momentum about point O.

Similarly and again, similarly again, you can look at

this in in any standard reference text for the development.

You can find that Euler's 2nd Law about the mass center, is

equal to the time derivative of the angular momentum about the Mass Center.

So here are the results, the moment equation about a point fixed in Inertial

Reference Frame about the Mass Center. However in general, you need to note

that this form of some of the moments equals the time derivative of the angular

momentum, only works for a point fixed in

a Inertial Reference Frame or the Mass Center.

It does not work for arbitrary point P.

And so in general, the sum of the moments about P does

not necessarily equal the time derivative of the angular momentum about P.

To show what that is, let's look at or recall

our knowledge of equivalent force systems from way back

in my my first course introduction to engineering mechanics.

We saw by a balance of moments that the sum

of the moments about P, balances with some of the moments about C plus R cross F.

So that's just a balance of equivalent force systems shown from

this picture.

Now in this case, in dynamics, we've developed that the sum of the moments

about C is equal to the time derivative of the angular momentum about C.

We know that force, the sum of the forces is the

time derivative of the linear momentum or d dt m vC.

And so we get the relationship for the sum of the moments about P is equal

to the time derivative of the angular momentum about C, plus rPC cross, m aC.

And so it's not in this form as shown up

here, but for an arbitrary point it's in this form.

So here's the summary.

Sum of the moments about O or C is equal to

time derivative about, of the angular momentum about O or C respectively.

That form is not true for an arbitrary

point, for an arbitrary point, this is, this

is true and we'll use this, these equations

quite a bit in the remainder of the course.

And so it's important to have a, a good understanding of those.

And I did want to make one other point. If you recall,

let's look at, we had the sum of the forces

equals the time derivative of the linear momentum for a body.

And so, what you'll see is, this is completely analogous.

[SOUND].

And so, this'll maybe give you a little bit better feel.

Sum of the forces is equal

to timed derivative of the linear momentums.

Sum of the moments is equal to the timed derivative of

the angular momentum, and so you see the, the connection there.

And that's where we'll pick up next time.

[BLANK_AUDIO]

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