General Physics Volume N 1

N1 – Newton’s Lawsand

N2 - Vector Calculus

General Physics Volume N 2

Newton’s First Law

• Total momentum of an isolated systemis conserved.

• In the absence of external interactions,an object’s or system’s center of massmoves at a constant velocity.

!

MvCM =r p 1

+r p 2

+ ...+r p N =

r p tot

General Physics Volume N 3

Newton’s Third Law

• A tiny impulse out of A is the sameas - flowing into A

• When objects A and B interact, theforce of the interaction exerts on A isequal in magnitude and opposite indirection to the force that it exerts on B.

!

r F B =

dr p [ ]

dt= " "

dr p [ ]

dt

#

$ %

&

' ( = "

r F A

!

dr p [ ]

!

dr p [ ]

General Physics Volume N 4

Two Minute Problem - N1T.3

• A large car drags a small trailer in such a waythat their common speed increases rapidly.Which tugs harder on the other?

• A. The car tugs harder on the trailer than viceversa.

• B. The trailer tugs harder on the car than viceversa.

• C. Both tug equally on each other• D. The trailer exerts no force on the car at all• E. There is no way to tell

General Physics Volume N 5

Two Minute Problem - N1T.4

• A parent pushes a small child on a swing so thatthe child moves rapidly away while the parentremains at rest. How does the magnitude of theforce that the child exerts on the parent compareto the magnitude of the force that the parentexerts on the child?

• A. The force on the child is larger• B. The force on the parent is larger• C. These forces have equal magnitudes• D. The child exerts zero force on the parent• E. There is no way to tell

General Physics Volume N 6

Two Minute Problem - N1T.6

• Arrows representing four forces having equal magnitudes areshown below. What combinations of these forces, actingtogether on the same object, will allow that object to move witha constant velocity?

• A. F2 and F3• B. F3 and F4• C. V1, F2, and F4• D. F1, F3, and F4• E. A and D• F. None of the above

!

r F 1

!

r F 2

!

r F 3

!

r F 4

!

r F 5

General Physics Volume N 7

Newton’s Second Law

• The next external force on an object causesits mass to accelerate

• One of the most important equations inphysics!

!

r F net,ext =

dr p tot

dt= M

dr v CM

dt" M

r a CM

General Physics Volume N 8

Classification of Forces

General Physics Volume N 9

Free-Body Diagrams

• Draw interaction diagram/conceptualdiagram

• Draw object alone• For each force, draw a dot where the

force is applied and an arrowrepresenting magnitude and direction

• Arrows only for forces• Every force arrow reflects an interaction

General Physics Volume N 10

Free-body diagram - book on a table

• Interaction/conceptual diagram• Draw object• Draw dot for each force• From dot draw direction of each force

General Physics Volume N 11

Vector Calculus

• Why?• In order to better understand velocity and

acceleration• Newton had to invent calculus to explain

his theory!

• Review in appendix NA and NB at endof Volume N

General Physics Volume N 12

• The time derivative is defined as

• The difference between and is that

• The first expression is at an instant in time• The second expression is for a short time interval

Time Derivative of a Vector

!

df

dt"

lim

#t$ 0

f t + #t( ) % f t( )#t

!

df

dt

!

"f

"t

General Physics Volume N 13

Derivative of a Vector

• The components of the time derivative vector are theordinary time derivatives of the components of thatvector

!

dr q

dt"

dqx

dtdqy

dtdqz

dt

#

$

% % % % % %

&

'

( ( ( ( ( (

General Physics Volume N 14

Definition of Velocity

• The velocity vector is the derivative ofthe displacement vector

!

r v "

dr r

dt"

small displacement

short time interval

!

r v "

dr r

dt"

dx

dtdy

dtdz

dt

#

$

% % % % %

&

'

( ( ( ( (

!

v = magr v ( ) = vx

2 + vy

2 + vz

2 =dx

dt

"

# $

%

& '

2

+dy

dt

"

# $

%

& '

2

+dz

dt

"

# $

%

& '

2

General Physics Volume N 15

Velocity - Instantaneous and Average

General Physics Volume N 16

Definition of Acceleration

• An object’s acceleration at an instant isa vector that expresses how rapidly andin what direction its velocity vector ischanging at that instant

!

r a "

dr v

dt"

dvx

dtdvy

dtdvz

dt

#

$

% % % % % %

&

'

( ( ( ( ( (

General Physics Volume N 17

Motion Diagrams

General Physics Volume N 18

Motion Map - Acceleration

General Physics Volume N 19

Two Minute Problem - N2T.4

• An object can have a constant speedand still be accelerating, true (T) orfalse (F)?

General Physics Volume N 20

Motion Diagram Example

General Physics Volume N 21

Two Minute Problem - N2T.5

• An object’s acceleration vectoralways points in the direction thatit is moving, true (T) or false (F)?

General Physics Volume N 22

Motion Map - 2D

General Physics Volume N 23

Uniform Circular Motion

• The magnitude of the acceration of anobject in uniform circular motion

• Depends on v and R• Points toward center of circle

!

a " magr a ( ) =

magr v ( )

2

R=

v2

R

General Physics Volume N 24

General Physics Volume N 25

Group ProblemsFree body diagrams

The ball is rising in aparabolic trajectory.

The object is pushed bya force applieddownward at an angle.

The object is pulled bya force parallel to thesurface.

Object slides withoutfriction.

Object slows due tokinetic friction.

Object slides atconstant speed withoutfriction.

General Physics Volume N 26

Group Activity - Make a Motion Map

• Throw the ball vertically upward• Not too high

• Using LoggerPro, capture the throw• On paper, draw a motion map• How does the velocity change?

• Draw velocity vectors

• Draw acceleration vectors

General Physics Volume N 27

Group Problems

• N1B.3• N1B.6• N1S.2• N2B.1• N2B.6• N2B.8