PHYSICS 361
(Classical Mechanics)
Dr. Anatoli Frishman
frishman@iastate.edu
Web Page: http://course.physastro.iastate.edu/phys361/
1
Introduction
•What is this course about
•A little bit of history
– Galileo (1564 – 1642)
– Newton (1642 – 1727)
– Lagrange (1736 – 1813)
– Hamilton (1805 – 1865)
– Poincare (1854 – 1912)
– Lyapunov (1857 – 1918)
– Lorenz (1917 – 2008)
Course organization
•Lectures
•Homework
•Exams (two midterm exams and a comprehensive final exam)
•Formula sheet
•Syllabus
2
1. Mechanics of material point
Definitions
Material point (point mass) – an object with an irrelevant
dimension for the purposes of a particular problem.
Position of material point in space:

r

Cartesian coordinates: r  ( x, y, z )  (rx , ry rz )  xiˆ  yˆj  zkˆ  xxˆ  yyˆ  zzˆ
xˆ  yˆ  zˆ  1
Velocity:
Acceleration:

 dr 
v
r
dt

2
 dv d r 
a
 2 r
dt dt
3
Newton’s laws
Newton’s First Law (the low of Inertia)
Existence of inertial systems of reference
In inertial system of reference, any object acted
by no net force remains at rest or continues its
motion along straight line with constant velocity
 dp
F
dt
Newton’s Second Law


p  mv
m  const 
Newton’s Third Law


F  ma


FAB   FBA
Note: these two forces
act on different objects.
Never add these forces!
4
Newton’s Second law as differential equation


F  ma

d r 1 
 F
2
dt
m
2
x  Fx m
Example (constant force): Fx  F  const 
Method 1:
v  x   xdt  C   F m dt  C  Ft m  C 
~
~
x   xdt  C   v0  at dt  C 
v  v0  at
at 2
x  x0  v0t 
2
Method 2:
v  v0  x  x0   xdx   F m dx  F t  t0  m
t
t
t0
t0
5
Example (Hook’s law):
Fx  kx
x  k m x
2  k m
x   2 x
This is a second-order differential equation.
The general solution contains two independents constants.
x  A sin t   
x  A cost   
x   A 2 sin t   
Another form of the solution:
x  B1 sin t   B2 cost 
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2. Polar and cylindrical coordinates
Definitions:
r  x 2  y 2
x  r cos  

y  r sin  
  arctan y x 
ˆ
ŷ
r̂
y

r r

ˆr  r r
x̂

r

x
Some properties:
xˆ  yˆ  rˆ  ˆ  1
rˆ  xˆ cos   yˆ sin 
ˆ   xˆ sin   yˆ cos 

F
A vector in polar and cylindrical coordinates coordinates:

Polar: F  Fr rˆ  Fˆ

Cylindrical: F  Fr rˆ  Fˆ  Fz zˆ
Fy
Fx
ˆ
r̂
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Derivatives in polar coordinates:
rˆ  xˆ cos   yˆ sin 
ˆ
   xˆ sin   yˆ cos 
rˆ   xˆ sin   yˆ cos  


ˆ  xˆ cos   yˆ sin  
ˆ(t )
dˆ
rˆ  ˆ

ˆ  rˆ
 
v  r  d rrˆ  dt  rrˆ  rrˆ 
d
ŷ
drˆ  d ˆ
d
rˆ(t )

x̂
rˆ  ˆ  1

v  rrˆ  rˆ
 

a  v  rrˆ  rrˆ  rˆ  rˆ  rˆ  rrˆ  2rˆ  rˆ  r 2 rˆ 

a  r  r 2 rˆ  r  2r ˆ

 




2





F

m
r

r


 r
F  ma  
 F  m r  2r

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Kinetic energy in polar coordinates:
T  12 mv 2
Method 1:
v x  x  d r cos   dt  r cos   r sin 
 v 2  v x2  v y2  r 2  r 2 2

v y  y  d r sin   dt  r sin   r cos 
Method 2:


2
2
2
2
ˆ

ˆ
rˆ    1  v  v   rrˆ  r  r 2  r 2 2
2
Polar:
Cylindrical:

mr
T  12 m r 2  r 2 2
T
1
2
2

 r 2 2  z 2

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Example (pendulum):
L

Ftension
r  L  const
Fr  mg cos   Ftension
F  m sin 
mg



 Fr  m r  r 2
mg cos   Ftension  mL 2


 mg sin   mL
 F  m r  2r

 2  g L    2 sin 
2







  1 
Ftension  mg cos   mL 2
The same equation as for the Hook’s law
  Asin t   
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