Summary for Chapters 24

Kinematics:

xv

t

Average velocity

0limt

xv

t

(Instantaneous) velocity

tan

Average acceleration =v

at

(Instantaneous) acceleration =0

limt

va

t

d v

d t

0

0 0

20 0

2 20 0

1

21

2

2

v v a t

x x v v t

x x v t a t

v v a x x

Constant acceleration:

3f x x

0

limx

d f x f x x f x

d x x

3 3

0limx

x x x

x

2 33 2 3

0

3 3limx

x x x x x x x

x

2 32

0

3 3limx

x x x x x

x

22

0lim 3 3x

x x x x

23x1

nnd x

n xd x

See Appendix A (page A-5) for a list of common derivatives.

Vectors

Vector = Quantity with magnitude & direction.

Free vectors: All vectors with the same magnitude & direction are equivalent.

C A B

Linear combination of vectors A & B :

, numbers

Linear independence : A & B are linearly independent if

0 A B 0

Vector space : Set of vectors that include every possible linear combinations among themselves.

Basis : Set of maximum number of independent vectors in the vector space.

Dimension : Number of vectors in the basis.

Every vector in the vector space can be represented as a linear combination of the basis vectors.

Vector Arithmetic with Unit Vectors

ˆ ˆx yA A A i j

ˆ ˆx yB B B i j

ˆ ˆx x y yA B A B A B i j

,x yA A

,x yB B

,x x y yA B A B

&x x y yA B A B A B

d

d t

va ˆ ˆ+ yx

d vd v

d t d t i j , yx

d vd v

d t d t

2 2

2 2,

d x d y

d t d t

2

2

d

d t

xd d

d t d t

x

Cartesian coordinate system:basis = rectilinear orthogonal unit vectors

2-D constant acceleration :

0

0

x x x

y y y

v v a t

v v a t

20 0

20 0

1

21

2

x x

y y

x x v t a t

y y v t a t

2

ˆv

ra r ( centripetal )

Uniform circular motion :

Projectile

2

0 0 0 02 20 0

tan2 cos

gy y x x x x

v

2r

Ta

Newton’s 1st law of motion (definition of inertia frame) :

Newton’s 2nd law of motion:

net

d

d t

pF

Newton’s 3rd law:

Normal force n : contact force acting normal to contact surface.

m in f = m a is the inertia mass (same everywhere).

Weight = force of gravity on mass:

mw g

0net const F v

m is gravitational mass

ab baF F

in inertial frame

The Fundamental Forces

The fundamental forces:

• Gravity: large scale phenomena

• Electroweak force

• Electromagnetic force: everyday phenomena

• Weak (nuclear) force

• Strong (nuclear) force 1

1025

1036

1038

Free Body Force Diagram

• Keep only features relevant to the dynamics.

• Replace all objects by points upon which forces act.

• Represent all forces by vectors.