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Oscillation motion
f
=
1
T
,
ω
=
2
π
T
SHM:
a
=
d
2
x
dt
2
=

ω
2
x
,
α
=
d
2
θ
dt
2
=

ω
2
θ
x
=
x
max
cos(
ω t
+
δ
),
x
max
=
A
v
=

v
max
sin(
ω t
+
δ
),
v
max
=
ω A
a
=

a
max
cos(
ω t
+
δ
) =

ω
2
x
,
a
max
=
ω
2
A
E
=
K
+
U
=
K
max
=
1
2
m
(
ω A
)
2
=
U
max
=
1
2
k A
2
Spring:
ma
=

k x
Simple pendulum:
ma
θ
=
mα‘
=

mg
sin
θ
Physical pendulum:
τ
=
I α
=

mg d
sin
θ
Torsion pendulum:
τ
=
I α
=

κθ
Gravity
~
F
21
=

G
m
1
m
2
r
2
12
ˆ
r
12
,
for
r
≥
R
,
g
(
r
) =
G
M
r
2
G
= 6
.
67259
×
10

11
Nm
2
/kg
2
R
earth
= 6370 km,
M
earth
= 5
.
98
×
10
24
kg
Circular orbit:
a
c
=
v
2
r
=
ω
2
r
=
‡
2
π
T
·
2
r
=
g
(
r
)
U
=

G
m M
r
,
E
=
U
+
K
=

G m M
2
r
F
=

d U
dr
=

mG
M
r
2
=

m
v
2
r
Kepler’s Laws of planetary motion:
i
) elliptical orbit,
r
=
r
0
1

²
cos
θ
r
1
=
r
0
1+
²
,
r
2
=
r
0
1

²
ii
)
L
=
r m
Δ
r
⊥
Δ
t
→
Δ
A
Δ
t
=
1
2
r
Δ
r
⊥
Δ
t
=
L
2
m
= const.
iii
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 Spring '08
 Kaplunovsky

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