Physics 303K course summary (11/24/99)
This sheet contains most of the basic formulas in Physics 303K. This sheet is intended to be a guide for
doing homework problems, for reviewing for the exams, and to be a reference for Physics 303L and for future
use. During any exam, questions on this sheet will not be answered. However, at other times your
questions are welcome.
Mathematics (§3, §7, §11, Appendix
p B)
•
For ax2 + bx + c = 0, x = (−b ± b2 − 4ac)/(2a).
•
Cartesian and polar coordinates:
Law of Motion and applications (§5,6)
x = r cos θ, y = r sin θ
•
Force: F~ = m~a, W = mg, F~12 = −F~21 .
2
r2 = x2 + y 2 , tan θ = xy
•
Circular motion: ac = vr , v = 2πr
•
T = 2πrf.
Trig : cos(α − β) = cos α cos β + sin α sin β
•
Friction: fs ≤ µs N, fk = µk N
α+β
α−β
sin α + sin β = 2 sin 2 cos 2
•
Gravity: F = Gmd12m2 . On earth : g = 9.8 m/s2 ,
α−β
cos α + cos β = 2 cos α+β
cos
•
2
2
Equilibrium (concurrent forces): Σi F~i = 0.
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ − sin2 θ
Energy (§7-8)
1 − cos θ = 2 sin2 θ2 , 1 + cos θ = 2 cos2 θ2 .
•
Work (for all F) :WA→B = F sk = Fk s = F s cos θ
•
~ = (Ax , Ay ) = Ax î + Ay ĵ
Vector algebra: A
RB
= F~ · ~s → A F~ · d~s (in Joules)
~ =A
~+B
~ = (Ax + Bx , Ay + By )
Resultant: R
•
Three types of effects due to work done:
Method of parallelgram: R is diagonal.
F ext = ma + F + f
• ~ ~
A · B = AB cos θ = Ax Bx + Ay By + Az Bz
W ext |A→B = (KB −KA )+(UB −UA )+Wdiss |A→B
•
Cross product: î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ.
RB
•
Kinetic energy: KB −KA = A F (x)dx, K = 12 mv 2
RB
î
•
ĵ
k̂ PE (conservative F~ ): UB − UA = − A F~ · d~s
~ =A
~×B
~ = Ax Ay Az •
C
Ugravity = mgy, Uspring = 21 kx2
Bx By Bz ∂U
•
From U to ~
F: Fx = − ∂U
∂x , Fy = − ∂y . . .
∂U
•
Fgravity = − ∂y = −mg, Fspring = − ∂U
∂x = −kx.
C = AB sin θ, = A B = AB , right hand rule.
⊥
dxn
dx
⊥
1
= nxn−1 , dlnx
dx = x
d cos θ
= − sin θ
dθ
Calculus:
= cos θ,
Measurements (§1)
•
SI units; Standards of length, mass, time.
•
Dimensional analysis: In any physics expression,
dimension of each item must be the same.
e.g.[F ] = [m][a] = M LT −2 , F = mi v j rk .
•
Signif. figures: c = ab, sf (c) = M in[sf (a), sf (b)]
PN
PN
•
Summation: i=1 (axi + b) = a i=1 xi + bN
Motion(§2,4)
dv
•
One dim motion: v = ds
dt , a = dt
sf −si
v −v
Average values: v̄ = tf −ti , ā = tff −tii
•
One dim motion with constant acceleration:
vt : v = v0 + at
st : s = v̄t = v0 t + 21 at2 , v̄ = v02+v .
vs : v 2 = v02 + 2as
•
Nonunif. acc.: Method of separation of variables.
•
The independent x and y motion
t
v0y
•
Projectile motion: trise = tf all = trip
2 = g
h = 12 gt2f all , R = vox ttrip
d sin θ
dθ
2
Circular: ac = vr , v = 2πr
, f = T1 (Hertz)
pT
Curvilinear motion: a = a2t + a2r .
•
Relative velocity: ~v = ~v 0 + ~u
2
•
∂ U
Equilibrium : ∂U
∂x = 0, ∂x2 > 0 (stable), < 0 (unst.)
•
~ v . (watts)
Power: P = dW
dt = F vk = F v cos θ = F ·~
Collision (§9)
Rt
•
Impulse :I~ = ∆~
p = p~f − p~i → tif F~ dt
•
Momentum : p~ = m~v
+m2 x2
•
Two-body,1 dim : xcm = m1 x1M
, M = m1 +m2
Pcm ≡ M vcm = p1 + p2
Fcm ≡ F1 + F2 = m1 a1 + m2 a2 = M acm
K1 + K2 = K1∗ + K2∗ + Kcm
•
Two-body collision: P~i = P~f = (m1 + m2 )~vcm
vi∗ = vi − vcm , vi0 = vi∗0 + vcm
•
Elastic: v1 − v2 = −(v10 − v20 ),
vi∗0 = −vi∗ , vi0 = 2vcm − vi
•
Many body, center
R of mass:
~rcm =
Σi mi ~
ri
M
~
r dm
= M , M = Σ i mi
~
•
Force on cm : F~ ext = ddtP = M~acm , P~ = Σi p~i
Rotation of Rigid-Body (§10)
•
Kinematics: θ = rs , ω = vr , α = artR.
•
Moment of inertia: I = Σi mi ri2 = r2 dm
Idisk = 21 M R2 , Iring = 12 M (R12 + R22 )
1
1
Irod = 12
M `2 , Irectangle = 12
M (a2 + b2 )
Isphere = 25 M R2 , Isph−shell = 23 M R2
I = M ∗ (Radius of gyration)2
I = Icm + M D2
•
Kinetic energies: Krot = 12 Iω 2 , K = Krot + Kcm Wave motion (§16)
P
•
•
Traveling waves: y = f (x − vt), y = f (x + vt).
Ang.momentum:L = rmv → i ri mi (ωri ) = Iω
Right
moving: y = A sin(kx − ωt − φ)
dL
dv
dω
•
Torque: τ = dt = m dt r = F r → I dt = Iα
q
•
W ext = ∆K+∆U +Wf , K = Krot + 21 mv 2 , P = τ ω • Along a string: v = Tµ
•
Rolling, angular momentum and torque (§11)
General: ∆E = ∆K + ∆U = ∆Kmax
1
1 Ic
•
2
2
2
Rolling: K = 2 (Ic + M R )ω = 2 ( R2 + M )v
•
1 2 2
•
~ = ~r × p~, L = r⊥ p → Iω.
Angular momentum: L
1 dim waves: P = ∆E
∆t = 2 A ω µv
~
d~
p
dL
•
∆m
∆m
∆A
∆m
~
Torque: ~τ = dt = ~r × dt = ~r × F , τ = r⊥ F → Iα. †• Circular: ∆t = ∆A · ∆r · ∆r
dt = ∆A · 2πrv
dφ
mgh
τ
•
∆m
†•
2
Gyroscope: ωp = dt = L1 dL
Spherical: ∆m
dt = L = Iω
∆t = ∆V · 4πr v
Static equilibrium (§12)
Sound q
(§17)
•
ΣF~i = 0, about any point Σ~τi = 0.
v = Bρ , s = smax cos(kx − ωt − φ)
mA ~
rAcm +mB ~
rBcm
•
Subdivisions: ~rcm =
∂s
mA +mB
∆P = −B ∆V
V = −B ∂x
•
Elastic modulus=stress/strain
∆Pmax = Bκsmax = ρvωsmax
∆m A∆x
•
stress: F/A
Piston: ∆m
∆t = ∆V · ∆t = ρAv
1
•
2
strain: ∆L/L, θ ≈ ∆x/h, −∆V /V .
Intensity: I = P
A = 2 ρv(ωsmax )
W
•
Oscillation motion (§13) f = T1 , ω = 2π
Intensity level: β = 10 log10 II0 , I0 = 10−12 m
2
T , λ = vT.
•
d2 x
d2 θ
•
2
2
Plane waves: ψ(x, t) = c sin(kx − ωt)
SHM : a = dt2 = −ω x, α = dt2 = −ω θ
†•
Circular waves: ψ(r, t) = √cr sin(kr − ωt)
x = xmax cos(ωt + δ) , xmax = A
†•
v = −vmax sin(ωt + δ) , vmax = ωA
Spherical: ψ(r, t) = rc sin(kr − ωt)
2
2
0
a = −amax cos(ωt + δ) = −ω x, amax = ω A
•
Doppler effect: λ = vT, f0 = T1 , f 0 = λv 0 .
1
1
•
2
2
E = K + U = Kmax = 2 m(ωA) = Umax = 2 kA .
Here v 0 = vsound ± vobserver , is wave speed
•
Spring : ma = −kx
relative to moving observer and
•
Simple pendulum : at = α` = −g sin θ
λ0 = (vsound ± vsource )/f0 , detected wave length
•
Physical pendulum: τ = Iα = −mgl sin θ
established by moving source of frequency f0 .
•
Torsion pendulum :τ = Iα = −κθ.
•
Received and reflected frequencies are the same.
Gravity (§14)
•
2
= sin1 θ
Shock waves: Mach Number= vvsource
R
•
sound
At r ≥ R : g(r) = GM
=
g
2
r
r
Superposition of waves (§18)
2
•
v2
2π
•
2
Phase difference: sin(kx − ωt) + sin(kx − ωt − φ)
Circular orbit: ac = r = ω r = T r = g(r)
•
Standing waves: sin(kx − ωt) + sin(kx + ωt)
2
GmM
mv
GmM
U = − GmM
, F = − dU
•
r
dr = − r 2 ,
r =
r2
Beats: sin(kx − ω1 t) + sin(kx − ω2 t)
G = 6.67 × 10−11 N m2 /kg 2 , Rearth = 6370 km .
†•
Others:
a cos(kx − ωt) + b sin(kx − ωt).
E = U + K = − GmM
2r .
y
=
sin(kx − ωt), z = sin(kx − ωt).
•
Kepler0 s Laws of planetary motion
•
Fundamental modes: Sketch wave patterns.
i. Elliptical orbit: Sun at one focal point.
Distance between adjacent nodes: λ/2.
r0
r0
0
√
r = 1−rcos
θ , r1 = 1+ r2 = 1− .
F/µ
λ
=
L;
f
=
String:
∆A
1 r∆r⊥
L
⊥
2
2L .
ii. L = rm ∆r
∆t → ∆t = 2 ∆t = 2m = const.
vs
2
2
Open-open pipe: λ2 = L; f = 2L
.
2πr
1
GM
2πa
1
=
→
=
.
iii. GM
vs
r2
T
r
a2
T
a
Open-closed pipe: λ4 = L; f = 4L
.
λ
Rod clamped at middle:
= L. Higher order
2
( a = r1 +r
= semimajor axis of the elliptic orbit.) modes: Sketch wave patterns. 2
2
•
Escape kinetic energy: E = K + U (R) = 0.
Two nodes with n antinodes between them: L =
Fluid mechanics (§15)
nλ/2.
q
•
Pascal: P = FA⊥1
= FA⊥2
, 1atm = 1.013×105 N/m2 .
n
T
1
2
String: nλ
2 = L; fn = 2L
µ , n = 1, 2, 3, . . ..
•
2
Archimedes: B = M g. Pascal = N/m .
nvs
•
Open-open pipe: nλ
P = Patm + ρgh, with P = FA⊥ and ρ = m
2 = L; fn = 2L , n = 1, 2, 3, . . ..
V .
nλ
R
R
s
h
Open-closed pipe: 4 = L; fn = nv
•
4L , n =
F = P dA → ρg` 0 (h − y)dy.
1, 3, 5, . . ..
•
Continuity equation: Av=constant.
Rod clamped at middle: nλ
•
2 = L, n = 1, 3, 5, . . ..
Bernoulli: P + 1 ρv 2 + ρgy=const. P ≥ 0.
2