soongsil universityclassical mechanics 2. newtonian mechanics newton ’ s law motion under constant...
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SOONGSIL UNIVERSITY
CLASSICAL MECHANICS
2. Newtonian Mechanics 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
2.2 Motion under a constant force
dtavv
dtavd
dt
vda
t
tv
v
0
0
00
200
0
0
0
0
2
1
)(0
tatvxx
dttavdtvxd
dt
xdv
tavv
ttx
x
SOONGSIL UNIVERSITY
CLASSICAL MECHANICS
• Bottom 에서의 속도는 ?• 마찰이 있을 때와 없을 때
• 마찰이 없을 때• 마찰이 없을 때 • 마찰이 있을 때• 마찰이 있을 때
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2.3 Position dependent force2.3 Position dependent force
• Potential E can be defined as• Potential E can be defined as
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2.3 Position dependent force continued2.3 Position dependent force continued
• Free fall
Maximum height of a ball thrown upwardMaximum height of a ball thrown upward
v0h
For a given initial velocity, there always exists maximum height.For a given initial velocity, there always exists maximum height.
Is it correct?Is it correct?
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2.3 Position dependent force continued2.3 Position dependent force continued
• Variation of Gravity with height• Variation of Gravity with height
• Escape speed• Escape speed
Definition of the gravitational accelerationDefinition of the gravitational acceleration
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2.3 Position dependent force continued2.3 Position dependent force continued
• Morse function V(x) : potential energy of a diatomic molecule• Morse function V(x) : potential energy of a diatomic molecule
Therefore, x=x0 is the equilibrium position.Therefore, x=x0 is the equilibrium position.
and –V0 is the binding energy.and –V0 is the binding energy.
x
separation of the atomsseparation of the atoms
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2.3 Position dependent force continued2.3 Position dependent force continued
• Near the equilibrium position V(x) can be expanded• Near the equilibrium position V(x) can be expanded
And it can be considered as a simple harmonic oscillator problem.And it can be considered as a simple harmonic oscillator problem.
The potential becomes parabolic in the leading order,The potential becomes parabolic in the leading order,
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Binding energy = -4.52 eVEquilibrium separation = 0.074 nmDelta = 0.036 nmRoom temp. ~ 1/40 eV
2.3 Position dependent force continued2.3 Position dependent force continued
• Hydrogen molecule• Hydrogen molecule
Maximum separation at room temp.?Maximum separation at room temp.?
X = 0.074 +- 0.0027 nm ~3.6% change in sizewhich means thermal expansion
X = 0.074 +- 0.0027 nm ~3.6% change in sizewhich means thermal expansion
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2.4 velocity dependent force2.4 velocity dependent force
The ratio
in SI unit,
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2.3 velocity dependent force continued2.3 velocity dependent force continued
As time goes to infinity, it converges to a point. As time goes to infinity, does it converge???
It shows logarithmic divergence!!!It shows logarithmic divergence!!!
• quadratic resistance• quadratic resistance (Dominant in high speed)• linear resistance• linear resistance (Dominant in low speed)
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• vertical fall through a fluid (linear resistance)• vertical fall through a fluid (linear resistance)
where the terminal speed = , and =
mg
-C1v
As time goes to infinity, the velocity approaches to –mg/c1.As time goes to infinity, the velocity approaches to –mg/c1. Terminal velocityTerminal velocity
It occurs when F = 0.It occurs when F = 0.
If an object is dropped, after 5 , v = 0.993 vtIf an object is dropped, after 5 , v = 0.993 vt
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• vertical fall through a fluid (quadratic resistance)• vertical fall through a fluid (quadratic resistance)
If an object is dropped, after 5 , v = 0.99991 vtIf an object is dropped, after 5 , v = 0.99991 vt
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• Terminal speed of raindrops and basketballs• Terminal speed of raindrops and basketballs
The ratio
Raindrop ~ 0.1 mmBasketball ~ 0.25 m
• raindrops• raindrops • basketballs• basketballs
The ratio = 0.14 v v = 7.1m/s
linear term dominates.linear term dominates.
The ratio = 350 v v = 0.0029m/s
quadratic term dominates.quadratic term dominates.
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The end
2. Newtonian Mechanics 2. Newtonian Mechanics
• Newton’s law• Motion under constant forces• Position dependent forces• Velocity dependent forces