Transcript the vector product - Tennessee State University

DYNAMICS
The motion of a body is affected by other bodies present in
the universe. This influence is called an interaction.
There are a few ways to describe this interaction by
mathematical models:
vectors:
• force
• torque
• impulse
scalars:
• work
• power
• heat
INERTIA
• Both the nature of the interaction and the characteristics of
the object determine the effect of the interaction on the
motion of the object.
• The “resistance” to change of motion is called inertia.
• Mass is a scalar quantity assigned to the inertial property
of a body
inertia
THE FIRST LAW OF MOTION
If a particle does not interact
with other bodies, it is possible
to find a reference frame in
which that particle has zero
acceleration.
Sir Isaac Newton (1642 - 1727)
(The 1729 translation by Andrew Motte from “Philosophiae Naturalis
Principia Mathematica”:
“Every body perseveres in its state of rest, or of uniform motion in a right line,
unless it is compelled to change that state by forces impressed thereon.”)
THE SECOND LAW OF MOTION
2
F41
1
F43
3



Fnet   Fi  ma
4
F42
a
i all
Fnet
In an inertial reference frame, the acceleration of a
particle is proportional to the net force (the sum of
all forces) exerted on the particle and inversely
proportional to the mass of the particle.
(“The alteration of motion is ever proportional to the motive force
impressed; and is made in the direction of the right line in which that force
is impressed.”)
lunar lander
THE THIRD LAW OF MOTION


F12  F21
F12
1
F21
2
If one body exerts a force on another body, the
second body exerts an opposite force on the first one.
(“To every action there is always opposed an equal reaction: or the mutual
actions of two bodies upon each other are always equal, and directed to
contrary parts.”)
FUNDAMENTAL FORCES
On the microscopic scale there are only four kinds of
interaction between particles:
• gravitational
• electromagnetic
• strong
• weak
GRAVITY
A particle with mass m1, separated from a
particle with mass m1 exerts an attractive
force (vector) on the other particle

F21   G m12m2  r12
r12
1
(Objects with spherically distributed mass,
also obey the above equations with r
representing the distance between the centers
of the objects. The force is exerted toward the
center of the object exerting the force.)
F21
r12
2
gravity
WEIGHT
The product of the mass of an
object and the free fall
acceleration at the location of the
object is called the weight of the
object:


W  mg
W
on earth g = 9.80 m/s2
On a planet with radius R and mass M the weight of a body is
approximately equal to the gravitational force exerted on the
object by the planet.

GM
W  m  2 rˆ
R
NORMAL FORCE
N
Fnet
W
The normal force is the vector
component of a force that a
rigid surface exerts (due to
deformation; strain) on an
object with which it is in
contact, namely, in the direction
perpendicular to the surface.
The normal force prevents objects from crossing the surface,
therefore it is dependent on the other forces applied to the body.
STATIC FRICTION
Static frictional force is a component of a force
that a rigid surface exerts (due to strain) on the
surface of an object with which it is in contact,
namely, in the direction parallel to the surface.
F
fs
N
Within certain limits the static
friction (interaction) prevents the
object's surface from moving along
the rigid surface.
f s  s N
W
KINETIC FRICTION
Kinetic frictional force is a component of a force that
a rigid surface exerts (due to deformation) on the
sliding surface of an object with which it is in
contact, namely, in the direction parallel to the
surface.
N
fk  k N
fk
f
Fnet
fs = kN
fs = -Fext
static
kinetic
Fext
W
friction
TENSION
Tension force is an interaction
that a surface exerts (due to
deformation; strain) on an
object with which it is in
contact,
namely,
in
the
direction perpendicular to the
surface.
T
Tension force prevents the object from leaving the
surface, therefore it is dependent on the other forces
applied to the body.
spring
Nicolaus Copernicus
1473-1543
2
T1
2
T3

2
R1
2
R3
How to weigh the earth?
Galileo Gallilei
1564-1642
Mm
FG 2
R
Johannes Kepler
1571-1630
Sir Isaac Newton
1642 - 1727
Henry Cavendish
1731-1810
inertial “forces”
position:
 z
R
r’



rt   Rt   r ' t 

(where r ' t   x' t ˆi ' t   y' t ˆj' t   z' t kˆ ' t  )
r’
y
O
x
velocity:





vt   Vt   v' t   t   r ' t 
acceleration:dR
' ˆ dz' ˆ 
dˆj'
dkˆ ' 
  dˆi '
the rate of change of the   vt      dx' ˆi' dy
 
j' k '    x '   y'  z'

dt
dt
dt
dt
dt
dt
dt



a  A  a'2  v'  r '    r '

d 

ˆj' v' kˆ ':   x ' ˆi ' y' ˆj'z' kˆ ' 
(primed)
at   V frame
 v' ˆi ' v' base




dt
 Vt   v' ˆi 'v' ˆj'v' kˆ '  x'   ˆi '  y'   ˆj'  z'   kˆ ' 

dV  dv' ˆ dv' ˆ dv' ˆ   dx'  ˆ dy'  ˆ dz'  ˆ 
de
ˆ    i' dt j' dt k '    dt   i'  dt   j' dtV t   kv''  ˆi'v' ˆj'v' kˆ '   x' ˆi'   y' ˆj'   z'kˆ ' 
dti '  dt
   eˆ i '
Newton’s
law
non-inertial
reference
  in


 frame:
dt  d  second
dx ' ˆa dy
' dz' 


x ' ˆi ' y' ˆj' z' kˆ '     
i ' ˆj' kˆ '   x '   ˆi '  y'   ˆj'  z'  kˆ ' 
dt
dt
  dt
dt  Vt  v' ˆi'v' ˆj'v' kˆ '  x' ˆi'y' ˆj'z'kˆ '
 
x
x
y
z
y
x
y
z
x
y
z
z
ma'  F
 mA  2m
  v'
 m  r ' m    r ' 
x
y
z



 
dV
d
net
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ

 a 'x i ' a 'y j' a 'z k '  2    v'x i ' v'y j' v'z k ' 
 x ' ˆi ' y' ˆj' z' kˆ '      x ' ˆi ' y' ˆj' z' kˆ '
dt
dt
