final 01 – RAMSEY, TAYLOR – Due: Dec 19 2006, 11:00 pm
1
Gravity
vector
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
=
parenleftBig
2
π
T
parenrightBig
2
r
=
g
(
r
)
U
=
−
G
mM
r
,
E
=
U
+
K
=
−
GmM
2
r
F
=
−
dU
dr
=
−
m G
M
r
2
=
−
m
v
2
r
Kepler’s Laws of planetary motion:
i
) elliptical orbit,
r
=
r
0
1
−
ǫ
cos
θ
r
1
=
r
0
1+
ǫ
,
r
2
=
r
0
1
−
ǫ
ii
)
L
=
r m
Δ
r
⊥
Δ
t
−→
Δ
A
Δ
t
=
1
2
r
Δ
r
⊥
Δ
t
=
L
2
m
= const.
iii
)
G
M
a
2
=
parenleftBig
2
π a
T
parenrightBig
2
1
a
,
a
=
r
1
+
r
2
2
,
T
2
=
parenleftBig
4
π
2
GM
parenrightBig
r
3
Escape kinetic energy:
E
=
K
+
U
(
R
) = 0
Fluid mechanics
Pascal:
P
=
F
⊥
1
A
1
=
F
⊥
2
A
2
,
1 atm = 1
.
013
×
10
5
N/m
2
Archimedes:
B
=
M g
,
Pascal=N/m
2
P
=
P
atm
+
ρ g h
,
with
P
=
F
⊥
A
and
ρ
=
m
V
F
=
integraltext
P dA
−→
ρ g ℓ
integraltext
h
0
(
h
−
y
)
dy
Continuity equation:
A v
= constant
Bernoulli:
P
+
1
2
ρ v
2
+
ρ g y
= const,
P
≥
0
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:
m a
=
−
k x
Simple pendulum:
m a
θ
=
m α ℓ
=
−
m g
sin
θ
Physical pendulum:
τ
=
I α
=
−
m g d
sin
θ
Torsion pendulum:
τ
=
I α
=
−
κ θ
Wave motion
Traveling waves:
y
=
f
(
x
−
v t
),
y
=
f
(
x
+
v t
)
In the positive
x
direction:
y
=
A
sin(
k x
−
ω t
−
φ
)
T
=
1
f
,
ω
=
2
π
T
,
k
=
2
π
λ
,
v
=
ω
k
=
λ
T
Along a string:
v
=
radicalBig
F
μ
Reflection of wave:
fixed end:
phase inversion
open end:
same phase
General:
Δ
E
= Δ
K
+ Δ
U
= Δ
K
max
P
=
Δ
E
Δ
t
=
1
2
Δ
m
Δ
t
(
ωA
)
2
Waves:
Δ
m
Δ
t
=
Δ
m
Δ
x
·
Δ
x
Δ
t
=
Δ
m
Δ
x
·
v
P
=
1
2
μ v
(
ω A
)
2
,
with
μ
=
Δ
m
Δ
x
Circular:
Δ
m
Δ
t
=
Δ
m
Δ
A
·
Δ
A
Δ
r
·
Δ
r
dt
=
Δ
m
Δ
A
·
2
π r v
Spherical:
Δ
m
Δ
t
=
Δ
m
Δ
V
·
4
π r
2
v
Sound
v
=
radicalBig
B
ρ
,
s
=
s
max
cos(
k x
−
ω t
−
φ
)
Δ
P
=
−
B
Δ
V
V
=
−
B
∂s
∂x
Δ
P
max
=
B κ s
max
=
ρ v ω s
max
Piston:
Δ
m
Δ
t
=
Δ
m
Δ
V
·
A
Δ
x
Δ
t
=
ρ A v
Intensity:
I
=
P
A
=
1
2
ρ v
(
ω s
max
)
2
Intensity level:
β
= 10log
10
I
I
0
,
I
0
= 10
−
12
W/m
2
Plane waves:
ψ
(
x, t
) =
c
sin(
k x
−
ω t
)
Circular waves:
ψ
(
r, t
) =
c
√
r
sin(
k r
−
ω t
)
Spherical:
ψ
(
r, t
) =
c
r
sin(
k r
−
ω t
)
Doppler effect:
λ
=
v T
,
f
0
=
1
T
,
f
′
=
v
′
λ
′
Here
v
′
=
v
sound
±
v
observer
, is wave speed relative
to moving observer and
λ
′
= (
v
sound
±
v
source
)
/f
0
,
detected wave length established by moving source of
frequency
f
0
.
f
received
=
f
reflected
Shock waves:
Mach Number=
v
source
v
sound
=
1
sin
θ
Superposition of waves
Phase difference:
sin(
k x
−
ωt
) + sin(
k x
−
ω t
−
φ
)
Standing waves:
sin(
k x
−
ω t
) + sin(
k x
+
ω t
)
Beats:
sin(
kx
−
ω
1
t
) + sin(
k x
−
ω
2
t
)
Fundamental modes:
Sketch wave patterns
String:
λ
2
=
ℓ
,
Rod clamped middle:
λ
2
=
ℓ
,
Open-open pipe:
λ
2
=
ℓ
,
Open-closed pipe:
λ
4
=
ℓ
Temperature and heat
Conversions:
F
=
9
5
C
+ 32
◦
,
K
=
C
+ 273
.
15
◦
Constant volume gas thermometer:
T
=
a P
+
b
Thermal expansion:
α
=
1
ℓ
d ℓ
dT
,
β
=
1
V
d V
dT
Δ
ℓ
=
α ℓ
Δ
T
,
Δ
A
= 2
α A
Δ
T
,
Δ
V
= 3
α V
Δ
T