rotation density and c of m 1 definition of a system energy of a system momentum of a system force a...

RotationDensity and C of M

1

Definition of a System

Energy of a System

Momentum of a System

Force a System

Work a System

Impulse on a System

Center-of-Mass


2

What is a System?

A system is the particle or group of particles as defined by a problem in physics.

It may be as small as a single atom consisting of neutrons, protons and electrons.

It may be as large as the entire universe.

It may or may not include every object in the problem.This depends on what is being asked?


3

The Work-Energy Theorem

The work-energy theorem is true for systems as well as for individual particles.

The work by all forces can be found using integration

The change in kinetic energy is just the sum of the change kinetic energy for each particle

systemsystem aon forcesby done KW

particlessystem KK

mass ofcenter systemon donesystem aon forcesby done rFW


4

The Work-Energy Theorem

Remember that for gravity and elastic forces, we can write

where

And so, we can write (remembering that there are other kinds of energy like energy of deformation, heat, etc.)

gravitygravityby done UW

elasticforce elasticby done UW

hmgU gravity

2elastic 2

1skU

totalothersystemsystemenergyfor with accountednot

system aon forcesby done EEUKW


5

Momentum and Collisions

There is nothing new here. You already learned that collisions required knowledge of systems and momenta add.


6

The Force on a System

The net force on a system is the sum of the net force on every particle in the system.

The particles to be considered in the system are given as part of the problem.

Example question: What is the net force on the system that includes the two books below?

particlesnetsystemnet FF

We only need to consider the forces ON each of the books.


7


The forces on the top book are:

1. Gravity from the earth2. Normal force from the apple3. Friction from the apple4. Normal force from the bottom book5. Friction from the bottom book

Example question: What is the net force on the system thatincludes the two books below?

The weight of the apple (gravity of the earth on the apple) is acting ON THE APPLE, not ON THE BOOK!!!


8


The forces on the bottom book are:

1. Gravity from the earth2. Normal force from top book3. Friction from the top book4. Normal force from the table5. Friction from the table6. Normal force from the hand


The weight of the top book (gravity of the earth on the top book) is acting ON THE TOP BOOK, not ON THE BOTTOM BOOK!!!


9


1. Gravity from the earth2. Normal force from the apple3. Friction from the apple4. Normal force from the bottom

book5. Friction from the bottom book


The force due to gravity on the system is the force of gravity on the top book plus the force of gravity on the bottom book. In other words, it is Mg.

1. Gravity from the earth2. Normal force from top

book1. Friction from the top book2. Normal force from the table3. Friction from the table4. Normal force from the hand

Top Book

Bottom Book


10


1. Gravity from the earth (on the system)2. Normal force from the apple3. Friction from the apple4. Normal force from the table5. Friction from the table6. Normal force from the hand


Look carefully. You will see that we could have treated both books as a single particle of mass M.

This is a general rule.

Thus, we are left with the following forces acting on the system (both books)…

System


11

The Work on a System

The net work on a system is the sum of the net work on every particle in the system.


Example question: What is the net work over a distance d on the system that includes the two books below?

net netsystem particlesW W



12


1. Gravity from the earth2. Normal force from the apple3. Friction from the apple4. Normal force from the bottom

book5. Friction from the bottom book


The force due to gravity on the system is the force of gravity on the top book plus the force of gravity on the bottom book. In other words, it is Mg.

1. Gravity from the earth2. Normal force from top

book1. Friction from the top book2. Normal force from the table3. Friction from the table4. Normal force from the hand

Top Book

Bottom Book


13




Look carefully. You will see that we could have treated both books as a single particle of mass M.

This is a general rule.


System


14

The Work on a System



We can now do the integral on the total force on the system. If this is difficult, we can do the integral separately for each force and add the results.


System


15

The Impulse on a System

The net impulse on a system is the sum of the net work on every particle in the system.


Example question: What is the net impulse over a time t on the system that includes the two books below?

net (system) net (particles)J J



16

The Impulse on a System


Example question: What is the net impulse over a time t on the system that includes the two books below?

We can now do the integral on the total force on the system. If this is difficult, we can do the integral separately for each force and add the results.


System


17

Definition of the center-of-mass

The center-of-mass of a system is that point in the system that follows the laws of physics for a single particle. In other words, it is the “position” of the system.

Motion of center-of-mass

The center-of-mass of any group of objects or any large single object follows the same principles as particles in the previous chapters.

Newton’s laws of motion still apply.

The constant acceleration equations still apply.

Work-energy theorem still applies.


18

Center-of-mass for individual particles

The center of mass of a system of objects is the point in space that behaves like a single point mass under the influence of external forces. It is the “position” of the system.

Center of mass for a group of point masses is given by the equation…

M

m

m

mN

i

ii

N

ii

N

iii

1

1

1cm

rrr


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Example for calculating center of mass

M

m

m

mN

i

ii

N

ii

N

iii

1

1

1cm

rrr

x

y

1 2

3 4kg 1

kg 2

kg 3

kg 2

4

3

2

1

m

m

m

m

kg 8M 2m

i m 2r

m 0r

j m 2i m 2r

j m 2r

4

3

2

1

j m 25.1i m 00.1kg 8

im 2kg 1m 0kg 2jm 2im 2kg 3jm 2kg 2r cm


20

Example: 2 spheres against One

1 2

3

F

F

1F

mg

Fn

fk

F*n

2F*n

mg

Fn

fk

3F

mg

Fn

fk

nk FmgFma *1 mgFma kn *2

mgFma k3

Things to notice

The acceleration a1 is not the same as a2.

The acceleration a1 is always smaller than a3.


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Example: 2 spheres against One

1 2

3

F

F

3F

mg

Fn

fk

gmFma k 22 cm

mgFma k3

Things to notice

For large forces, acm is smaller than a3.

For small forces, acm is almost equal to a3.

2F

2mg

Fn

fk

1


22

Velocity and Acceleration of Center-of-Mass

dt

rd

M

vmv

N

iii

cm1cm

dt

vd

M

ama

N

iii

cm1cm

rotation density and c of m 1 definition of a system energy of a system momentum of a system force a...

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force of gravity

net force

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