Transcript soongsil university

2. Newtonian Mechanics
•
•
•
•
Newton’s law
Motion under constant forces
Position dependent forces
Velocity dependent forces
SOONGSIL UNIVERSITY
CLASSICAL MECHANICS
2.2 Motion under a constant force

 dv
a
dt

v
t

v0
0


d
v

a
  dt
t
 

v  v0   adt
0
SOONGSIL UNIVERSITY
  
v  v0  at

 dx
v
dt

x
t
t


 
 dx   vdt   (v0  at )dt

x0
0
0
  
12
x  x0  v0t  at
2
CLASSICAL MECHANICS
• Bottom 에서의 속도는?
• 마찰이 있을 때와 없을 때
• 마찰이 없을 때
SOONGSIL UNIVERSITY
• 마찰이 있을 때
CLASSICAL MECHANICS
2.3 Position dependent force
• Potential E can be defined as
SOONGSIL UNIVERSITY
CLASSICAL MECHANICS
2.3 Position dependent force continued
• Free fall
Maximum height of a ball thrown upward
h
v0
For a given initial velocity, there always exists maximum height.
Is it correct?
SOONGSIL UNIVERSITY
CLASSICAL MECHANICS
2.3 Position dependent force continued
• Variation of Gravity with height
Definition of the gravitational acceleration
• Escape speed
SOONGSIL UNIVERSITY
CLASSICAL MECHANICS
2.3 Position dependent force continued
• Morse function V(x) : potential energy of a diatomic molecule
x

Therefore, x=x0 is the equilibrium position.

separation of the atoms
and –V0 is the binding energy.
SOONGSIL UNIVERSITY
CLASSICAL MECHANICS
2.3 Position dependent force continued
• Near the equilibrium position V(x) can be expanded
The potential becomes parabolic in the leading order,
And it can be considered as a simple harmonic oscillator problem.
SOONGSIL UNIVERSITY
CLASSICAL MECHANICS
2.3 Position dependent force continued
• Hydrogen molecule
Maximum separation at room temp.?
Binding energy = -4.52 eV
Equilibrium separation = 0.074 nm
Delta = 0.036 nm
Room temp. ~ 1/40 eV
X = 0.074 +- 0.0027 nm
~3.6% change in size
which means thermal expansion
SOONGSIL UNIVERSITY
CLASSICAL MECHANICS
2.4 velocity dependent force
in SI unit,
The ratio
SOONGSIL UNIVERSITY
CLASSICAL MECHANICS
2.3 velocity dependent force continued
• linear resistance
(Dominant in low speed)
As time goes to infinity, it converges to a point.
• quadratic resistance
(Dominant in high speed)
As time goes to infinity, does it converge???
It shows logarithmic divergence!!!
SOONGSIL UNIVERSITY
CLASSICAL MECHANICS
• vertical fall through a fluid (linear resistance)
-C1v
mg
As time goes to infinity, the velocity approaches to –mg/c1.
Terminal velocity
It occurs when F = 0.
where the terminal speed
=
If an object is dropped, after 5
SOONGSIL UNIVERSITY
, and
=
, v = 0.993 vt
CLASSICAL MECHANICS
• vertical fall through a fluid (quadratic resistance)
If an object is dropped, after 5
SOONGSIL UNIVERSITY
, v = 0.99991 vt
CLASSICAL MECHANICS
• Terminal speed of raindrops and basketballs
Raindrop ~ 0.1 mm
Basketball ~ 0.25 m
The ratio
• raindrops
The ratio = 0.14 v
• basketballs
v = 7.1m/s
linear term dominates.
SOONGSIL UNIVERSITY
The ratio = 350 v
v = 0.0029m/s
quadratic term dominates.
CLASSICAL MECHANICS
2. Newtonian Mechanics
•
•
•
•
Newton’s law
Motion under constant forces
Position dependent forces
Velocity dependent forces
The end
SOONGSIL UNIVERSITY
CLASSICAL MECHANICS