Elements of classical mechanics - mvputten.org · I. Trajectories Cartesian and polar ... II....
Transcript of Elements of classical mechanics - mvputten.org · I. Trajectories Cartesian and polar ... II....
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Contents
I. Trajectories
Cartesian and polar coordinates
Rotations
Energy and forces
II. Hyperbolic and elliptic trajectories
III. Three classical mechanics problems
Hooke’s spring
Newton’s apple
The jumper
Particle trajectories
Cartesian and polar coordinates
Trajectories: bound and unbound
Energy and forces
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Basis vectors
A=(a,b)
x-axis
y-axis
b
a
i
j
{i,j} = orthonormal basis:
i,j have unit length
i,j are orthogonala = projection of A onto the x-axis
b = projection of A onto the y-axis
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Representations of a vector
A = a i + b j = a 10
⎛
⎝⎜⎞
⎠⎟+ b 0
1⎛
⎝⎜⎞
⎠⎟
= a0
⎛
⎝⎜⎞
⎠⎟+ 0
b⎛
⎝⎜⎞
⎠⎟= a
b⎛
⎝⎜⎞
⎠⎟
i = 10
⎛
⎝⎜⎞
⎠⎟, j = 0
1⎛
⎝⎜⎞
⎠⎟
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choice of basis vectors
Rotation of a coordinate system
x-axis
y-axis
b’ a’
Ay’-axis
x’-axis
{i,j} {i’,j’}
ϕ
Rotation of an ONB
In rotating a coordinate system, vectors A remain unchanged
a’ = projection of A onto the x’-axis
b’ = projection of A onto the y’-axis
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— projections
{i’, j’} from {i,j}:
a’ = projection of X onto the x’-axis
b’ = projection of X onto the y’-axis
x-axis
y-axis
Xunit circle
1
sinϕ
cosϕ
ϕi ' = cosϕ i + sinϕ jj ' = −sinϕ i + cosϕ j
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Rotation: a linear transformation
A = a 'i '+ b ' j '= a '(cosϕ i + sinϕ j) + b'(−sinϕ i + cosϕ j)= (a 'cosϕ − b 'sinϕ )i +(a'sinϕ + b 'cosϕ ) j≡ ai +bj
ab
⎛
⎝⎜⎞
⎠⎟= R a '
b '⎛
⎝⎜⎞
⎠⎟=
cosϕ −sinϕsinϕ cosϕ
⎛
⎝⎜
⎞
⎠⎟
a 'b '
⎛
⎝⎜⎞
⎠⎟
R = rotation matrix
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— example (I)
ϕ =ωt
ab
⎛
⎝⎜⎞
⎠⎟=
cosωt −sinωtsinωt cosωt
⎛
⎝⎜⎞
⎠⎟a 'b '
⎛
⎝⎜⎞
⎠⎟
In a co-rotating frame of the moon:
ω =2πP
angular velocity of circular motion with period P
Earth
Moon
a 'b '
⎛
⎝⎜⎞
⎠⎟= R
0⎛
⎝⎜⎞
⎠⎟
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:
— example (II)ab
⎛
⎝⎜⎞
⎠⎟= R cosωt −sinωt
sinωt cosωt⎛
⎝⎜⎞
⎠⎟10
⎛
⎝⎜⎞
⎠⎟
= R cosωtsinωt
⎛
⎝⎜⎞
⎠⎟ Earth
Moon
A = ab
⎛
⎝⎜⎞
⎠⎟,
dAdt
= da / dtdb / dt
⎛
⎝⎜⎞
⎠⎟= Rω −sinωt
cosωt⎛
⎝⎜⎞
⎠⎟
Earth
Moon
A
dA/dtvelocity vector
position vector
dAdt
=dadt
⎛⎝⎜
⎞⎠⎟
2
+dbdt
⎛⎝⎜
⎞⎠⎟
2
= Rω
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||A||=R d||A||/dt=0
:
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ddti ' =ω j '
ddtj ' = −ω i '
⎧
⎨⎪⎪
⎩⎪⎪
— example (III)
{i '(t), j '(t)}Same applied to the rotating ONB
i’(t)
i’(t+dt) di’j’(t)
j’(t+dt)
dj’
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dϕdt
=ω
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Trajectories in Newton’s gravitational field
Bound orbits
Unbound trajectories
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Trajectories in Newtonian gravity
Bound: elliptical orbits
Unbound: hyperbolic orbitsfocal points at ± p
ϕ
m
M
−ϕ0 <ϕ(t) <ϕ0
p-p
M
ml1 l2l1 + l2 = const.
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Elliptical trajectories (Kepler)
p-p
M
ml1 l2
l1 + l2 = const.
l1,2 = (x ± p)2 + y2
(l2 − 4 p2 )x2 + l2y2 = 14l2 (l2 − 4 p2 )
xa
⎛⎝⎜
⎞⎠⎟
2
+yb⎛⎝⎜
⎞⎠⎟
2
= 1
semi-major axis a semi-minor axis b
For a > b:
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Hyperbolic trajectories (“scattering”)
ϕ
m
M
−ϕ0 <ϕ(t) <ϕ0
r = rcosϕsinϕ
⎛
⎝⎜
⎞
⎠⎟ , r = r ϕ( )Polar coordinates:
Newton’s gravitational force F = −GMmr2
r̂, r̂ = rr,
Can show: r = const.sinϕ0 + sinϕ
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Open and closed according to total energy H
Bound orbits:
Unbound trajectories:
closed
open (reach out to infinity)
H < 0
H > 0
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Kinetic and potential energy
Kinetic energy:
Potential energy (Newtonian gravitational binding energy):
Ek =12mv2
U = −GMmr
v
r
Ek ∝ v2 ≥ 0
U ∝−Mmr
≤ 0
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Total energy H
Total energy H: kinetic energy plus potential energy
H = Ek +U =12mv2 − GMm
r
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One-dimensional “orbits”
v = drdt
m dvdt= F = − d
drU = −
GMmr2
M m
linear motion in one dimension
dHdt
= mv dvdt+GMmr2
drdt= v m dv
dt+GMmr2
⎛⎝⎜
⎞⎠⎟= 0
Newton’s law of gravity
Total energy H is conserved
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Classification of orbits by H
H = Ek +U =12mv2 − GMm
r
Since Ek ≥ 0,U→ 0
H < 0H > 0
bound orbits: forbidden to reach infinity
upon approaching infinity
unbound orbits: allowed to reach infinity
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“Black objects”
1793: John Michell
1795: Pierre Laplace
1915: Karl Schwarzschild (exact solution)
1967: John Wheeler’s “black hole”
1974: Stephen Hawking: black holes are grey, emitting thermal radiation
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“Black objects”
H = 0 : 12mve
2 −GMmR
= 0
ve : initial velocity at the surface (“escape velocity”)
H<0: fall back
H>0: successful escapeH=0: “trapped surface” within which H < 0
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🙂
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Black holes
12mc2 − GMm
RS= 0 :
ve = cConsider the limit (velocity of light)
RS =2GMc2
R = RS Radius of a
Schwarzschild black hole
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When does Newton’s law of gravity apply?
Newton’s law of gravitational attraction to a mass M applies, provided that
a) distances >> Schwarzschild radius of M
b) accelerations >> cosmological background acceleration (defined by the velocity of light and the Hubble parameter)
Newton’s law applies very well to the solar system, except for small deviations in the orbit of Mercury (small but important!)
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Work and potential energy
ropespring
m pull
wall
a) Fs = −Fr
Fs Fr
Newton’s third law
b)Work performed stored in spring potential energyΔEs =W
W
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Work and potential energy
ΔEs = − Fs0
Δl
∫ ds
W = Fr0
Δl
∫ ds
Newton’s third law ΔEs =W
Δl
Changes are assumed to be slow and conservative, neglecting kinetic energy in m and dissipation by friction
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— example: linear spring
Fs = −kl
k is spring constant:
length of spring
force on the spring
linear range
over-stretched, nonlinear
[k]= [F][l]
=g cms−2
cm= g s−2
Hooke’s law (1660):
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— Aside: Young’s modulus
Formulation of Hooke’s law in dimensionless strain ΔL/L
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— strain-stress correlation
Young: linear relationship between stress and strain
solid material
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F = Aσ
σ = E Δll
E is Young’s modulus
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— strain-stress correlation
dimensionless strain
stress linear range
over-stretched, nonlinear
[E]= [F / A][Δl / l]
= g cm−1s−2
solid material
E large: stiff material E small: soft material
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Dimensional analysis:
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Hooke’s pendulum clock
u
Hooke: linear relationship between force and stretch
length l
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F = ku
u = l0 − l
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Hooke’s pendulum clock
u
u(t)t
F0 = mg = kl0Gravitational force balanced by a stretch of the spring:
length l
l = l0 − uΔF = F − F0 = −k(l − l0 ) = −ku
ΔF = ma = −m&&uNewton’s third law
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Ü
m&&u = −kuÜ
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Hooke’s pendulum clock
u
u(t)t
Time-harmonic deflection
length l
u = Acos ωt +ϕ0( ),
Equation of motion
is a 2nd order ordinary differential equation:
Two integration constants: amplitude A, initial phase ϕ0
ω =km,
P = 2πω
= 2π mk
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m&&u = −kuÜ
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— trajectory in space and time
height u(t)
Free fall initial value problem
velocity du(t)/dt
time tT
fall (drop) =
area A(t)
area A(t)
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m&&u = −mgu(0) = H&u(0) = 0
⎧
⎨⎪
⎩⎪
Ü
ů(0)ů(0)
mů(t)=-mg
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— free fall time
Integrate:
Integrate a second time:
u(t)− u(0) = − 12gt 2 : u(t) = H −
12gt 2
u(T ) = 0 : T =2HgFree fall time:
(A(t) = ½ g t2) u(t) = H – A
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&u(t)− &u(0) = −gt : &u(t) = −gtů(t) - ů(0) ů(t)
(“H - area”)
ů(t)
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m&&u = −mgu(0) = 0&u(0) =V
⎧
⎨⎪
⎩⎪
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— trajectory in space and time
The jumper “flies” according to the initial value problem
with prescribed initial height and velocity
velocity du(t)/dt
T
area A(t)
height u(t)
time tjump height area A(t)
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ů(0) ů(0) mů(t)=-mg
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— total flight time
Integrate once
&u(t)− &u(0) = −gt : &u(t) =V − gtIntegrate a second time
u(t)− u(0) =Vt − 12gt 2 : u(t) =Vt − 1
2gt 2
u(T ) = 0 : T =2VgTotal flight time
&u(t*) = 0 : t* =
Vg
Time at maximal height
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ů(t) - ů(0) ů(t)
ů(t)
ů(t)
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— jump height
u(t) =Vt − 12gt 2 =VT y τ( ), y τ( ) = τ 1−τ( )
Parabolic trajectory
τ =tT
expressed in dimensionless time
For parabolic curves: max y τ( ) = 14
τ =12
⎛⎝⎜
⎞⎠⎟
Maximal height reached:14VT =
V 2
2g
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=Ekmg
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— duration of flying high
y(τ ) = 18 y = τ (1−τ ) = 1
4− τ −
12
⎛⎝⎜
⎞⎠⎟
2
18= τ −
12
⎛⎝⎜
⎞⎠⎟
2
: τ ± =12±
12 2
Solve
t+ − t−T
= τ + −τ− =12≅ 0.71
Relative time of flight above one-half the maximum height is
1/4
1/8
u(t)
t
H
H/2
y(τ )
τ
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— polar coordinates
The trajectory of A is an ellipse: r + s = c (c is some constant)
-p px-axis
oϕ γ
A
r sr sinϕ = ssinγr cosϕ = 2p + l cosγ⎧⎨⎩
r2 = s2 + 4 pscosγ + 4 p2
scosγ = r cosϕ − 2ps = c − r
⎧
⎨⎪
⎩⎪
Write out
r(ϕ ) =
12c − 2p
2
c
1− 2p2
ccosϕ
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