Mechanics - Basic Physical Concepts
Mathematics
√
Quadratic Eq.: a x2 + b x + c = 0, x = −b± 2ba−4 a c
Cartesian and polar coordinates:
y
x = r cos θ, y = r sin θ, r 2 = x2 + y 2 , tan θ = x
Trigonometry:
cos α cos β + sin α sin β = cos(α − β)
α−β
sin α + sin β = 2 sin α+β
2 cos 2
2
α−β
cos α + cos β = 2 cos α+β
2 cos 2
sin 2 θ = 2 sin θ cos θ, cos 2 θ = cos2 θ − sin2 θ
1 − cos θ = 2 sin2 2θ , 1 + cos θ = 2 cos2 2θ
~ = (Ax , Ay ) = Ax ı̂ + Ay ̂
Vector algebra: A
~
~+B
~ = (Ax + Bx , Ay + By )
Resultant:
R=A
~·B
~ = A B cos θ = Ax Bx + Ay By + Az Bz
Dot: A
Cross product: ı̂ × ̂ = k̂, ̂ × k̂ = ı̂, k̂ × ı̂ = ̂
¯
¯
¯ ı̂
̂
k̂ ¯
¯
¯
~ =A
~×B
~ = ¯ Ax Ay Az ¯
C
¯
¯
¯ B
¯
x By Bz
C = A B sin θ = A⊥ B = A B⊥ , use right hand rule
d xn = n xn−1 ,
d
1
Calculus: dx
dx ln x = x ,
d
d
d
dθ sin θ = cos θ, dθ cos θ = − sin θ, dx const = 0
Measurements
Dimensional analysis: e.g.,
2
F = m a → [M ][L][T ]−2 , or F = m vr → [M ][L][T ]−2
PN
PN
Summation:
i=1 (a xi + b) = a i=1 xi + b N
Motion
One dimensional motion: v = ddts , a = ddtv
s −s
v −v
Average values: v̄ = tff −tii , ā = tff −tii
One dimensional motion (constant acceleration):
v(t) : v = v0 + a t
s(t) : s = v̄ t = v0 t + 12 a t2 , v̄ = v02+v
v(s) : v 2 = v02 + 2 a s
Nonuniform acceleration: x = x0 + v0 t + 12 a t2 +
1 j t3 + 1 s t4 + 1 k t5 + 1 p t6 + . . ., (jerk, snap,. . .)
6
24
120
720
Projectile motion: trise = tf all =
h = 12 g t2f all , R = vox ttrip
ttrip
v0y
2 = g
2
Circular: ac = vr ,
v = 2 Tπ r , f = T1 (Hertz=s−1 )
q
Curvilinear motion: a = a2t + a2r
Relative velocity: ~v = ~v 0 + ~u
Law of Motion and applications
Force: F~ = m ~a, Fg = m g, F~12 = −F~21
2
Circular motion: ac = vr , v = 2 Tπ r = 2 π r f
Friction: Fstatic ≤ µs N
Fkinetic = µk N
P ~
Equilibrium (concurrent forces):
i Fi = 0
Energy
Work (for all F): ∆W = WA→B = WB − WA =
RB
F sk = Fk s = F s cos θ = F~ · ~s → A
F~ · d~s (in Joules)
Effects due to work done: Fext = m a + Fc + fnc
Wext |A→B = KB − KA + UB − UA + Wdiss |A→B
RB
Kinetic energy: KB −KA = A
m ~a ·d~s, K = 12 m v 2
R
K (conservative F~ ): U − U = − B F~ · d~s
B
A
A
Ugravity = m g y,
Uspring = 12 k x2
U , F = −∂ U , F = −∂ U
From U to F~ : Fx = − ∂∂x
y
z
∂y
∂z
U = −m g,
Fgravity = − ∂∂y
U = 0,
Equilibrium: ∂∂x
U = −k x
Fspring = − ∂∂x
∂ 2 U > 0 stable, < 0 unstable
∂x2
W = F v = F v cos θ = F
~ · ~v (Watts)
Power: P = ddt
k
Collision
Rt
Impulse: I~ = ∆~
p = p~f − p~i → tif F~ dt
Momentum: p~ = m ~v
x1 +m2 x2
Two-body: xcm = m1m
1 +m2
pcm ≡ M vcm = p1 + p2 = m1 v1 + m2 v2
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 = 2 vcm − vi
R
P
~r dm
m ~r
Many body center of mass: ~rcm = P i i = R
mi
p
Force on cm: F~ext = d~
dt = M~acm ,
p~ =
Rotation of Rigid-Body
mi
P
p~i
Kinematics: θ = rs , ω = vr , α = art
R
P
Moment of inertia: I =
mi ri2 = r2 dm
Idisk = 12 M R2 , Iring = 12 M (R12 + R22 )
1 M `2 , I
1
2
2
Irod = 12
rectangle = 12 M (a + b )
Isphere = 25 M R2 , Ispherical shell = 23 M R2
I = M (Radius of gyration)2 , I = Icm + M D2
Kinetic energies: Krot = 12 I ω 2 , K = Krot + Kcm
Angular momentum: L = r m v = r m ω r = I ω
Torque: τ = ddtL = m ddtv r = F r = I ddtω = I α
Wext = ∆K +∆U +Wf ,
K = Krot + 12 m v 2 ,
P =τω
Rolling, angular
momentum
and
³
´
³ torque
´
Ic + M v 2
Rolling: K = 12 Ic + M R2 ω 2 = 12 R
2
~
Angular momentum: L = ~r × p~, L = r⊥ p = I ω
~
Torque: ~τ = d L = ~r × d~p = ~r × F~ , τ = r F = I α
dt
dt
1 dL = τ = mgh
Gyroscope: ωp = ddtφ = L
L
Iω
dt
Static equilibrium
P
P~
~τi = 0
Fi = 0, about any point
+mB ~rBcm
Subdivisions: ~rcm = mA ~rAcm
mA +mB
Elastic modulus = stress/strain
stress: F/A
strain: ∆L/L, θ ≈ ∆x/h, −∆V /V
⊥
Gravity
F~21 = −G m12m2 r̂12 ,
r12
for r ≥ R,
g(r) = G M
r2
G = 6.67259 × 10−11 N m2 /kg2
Rearth = 6370 km, Mearth = 5.98 × 1024 kg
³ ´2
2
Circular orbit: ac = vr = ω 2 r = 2Tπ r = g(r)
U = −G mrM ,
M
E = U + K = −Gm
2r
Spherical: ψ(r, t) = rc sin(k r − ω t)
r0
r2 = 1−²
L
⊥
⊥
−→ ∆A = 12 r ∆r
const.
ii) L = r m ∆r
∆t = 2 m =
³ 2´
³∆t ´2 ∆t
r
+r
1
M
2
π
a
4π
2
3
1
2
iii) G a2 =
a, a =
2 , T = GM r
T
Escape kinetic energy: E = K + U (R) = 0
Fluid mechanics
Pascal: P = FA⊥1 = FA⊥2 , 1 atm = 1.013 × 105 N/m2
1
2
Archimedes: B = M g, Pascal=N/m2
P = Patm + ρ g h, with P = FA⊥ and ρ = m
V
R
R
F = P dA −→ ρ g ` 0h (h − y) dy
Continuity equation: A v = constant
P ≥0
Bernoulli: P + 12 ρ v 2 + ρ g y = const,
Oscillation motion
f = T1 , ω = 2Tπ
2
2
S H M: a = ddt2x = −ω 2 x, α = ddt2θ = −ω 2 θ
x = xmax cos(ω t + δ), xmax = A
v = −vmax sin(ω t + δ), vmax = ω A
a = −amax cos(ω t + δ) = −ω 2 x, amax = ω 2 A
E = K + U = Kmax = 12 m (ω A)2 = Umax = 12 k A2
Spring: m a = −k x
Simple pendulum: m aθ = m α ` = −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)
λ
T = f1 , ω = 2Tπ , k = 2λπ , v = ω
k = T
q
Along a string: v = F
µ
fixed end: phase inversion
open end: same phase
General: ∆E = ∆K + ∆U = ∆Kmax
1 ∆m
2
P = ∆E
∆t = 2 ∆t (ωA)
∆m ∆x
∆m
Waves: ∆m
∆t = ∆x · ∆t = ∆x · v
P = 12 µ v (ω A)2 , with µ = ∆m
∆x
∆m ∆A ∆r
∆m
Circular: ∆m
∆t = ∆A · ∆r · dt = ∆A · 2 π r v
∆m
2
Spherical: ∆m
∆t = ∆V · 4 π r v
Reflection of wave:
v=
q
Sound
B,
ρ
s = smax cos(k x − ω t − φ)
1
2
Intensity: I = P
A = 2 ρ v (ω smax )
I
Intensity level: β = 10 log10 I , I0 = 10−12 W/m2
0
Plane waves: ψ(x, t) = c sin(k x − ω t)
Circular waves: ψ(r, t) = √c sin(k r − ω t)
r
2
F = − ddrU = −m G M
= −m vr
r2
Kepler’s Laws of planetary motion:
r0
0
i) elliptical orbit, r = 1−²rcos
θ r1 = 1+² ,
∂s
∆P = −B ∆V
V = −B ∂x
∆Pmax = B κ smax = ρ v ω smax
∆m A ∆x
Piston: ∆m
∆t = ∆V · ∆t = ρ A v
0
v
Doppler effect: λ = v T , f0 = T1 , f 0 = λ
0
0
Here v = vsound ± vobserver , is wave speed relative
to moving observer and λ0 = (vsound ± vsource )/f0 ,
detected wave length established by moving source of
frequency f0 . freceived = fref lected
Shock waves: Mach Number= vvsource
= sin1 θ
sound
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 − ω1 t) + sin(k x − ω2 t)
Fundamental modes: Sketch wave patterns
λ
String: λ
2 = `, Rod clamped middle: 2 = `,
Open-open pipe: λ
2 = `,
Open-closed pipe: λ
4 =`
Temperature and heat
Conversions: F = 95 C + 32◦ ,
K = C + 273.15◦
Constant volume gas thermometer: T = a P + b
Thermal expansion: α = 1` ddT` , β = V1 ddTV
∆` = α ` ∆T , ∆A = 2 α A ∆T , ∆V = 3 α V ∆T
Ideal gas law:
P V = nRT = N kT
R = 8.314510 J/mol/K = 0.0821 L atm/mol/K
k = 1.38 × 10−23 J/K, NA = 6.02 × 1023 , 1 cal=4.19 J
Calorimetry: ∆Q = c m ∆T, ∆Q = L ∆m
R
First law: ∆U = ∆Q − ∆W , W = P dV
∆T
−H `i
Conduction: H = ∆Q
∆t = −k A ∆` , ∆Ti = A ki
Stefan’s law: P = σ A e T 4 , σ = 5.67 × 10−8 m2WK 4
Kinetic theory of gas
2
m vx
x
F = ∆p
∆t = d
N 2
Pressure: P = NAF = mVN vx2 = m
3V v
K
1
P = 23 N
V K, K x = 3 = 2 k T , T = 273 + Tc ,
P V = N k T , n = N/NA , k = 1.38×10−23 J/K,
NA = 6.02214199 × 1023 #/kg/mole
Constant V:
∆Q = ∆U = n CV ∆T
Constant P:
∆Q = n CP ∆T
Ideal gas: ∆px = 2 m vx ,
C
γ = C P , CP − C V = R
V
CV = d2 R, for transl.+rot+vib, d = 3 + 2 + 2
Adiabatic expansion: P V γ = constant
t
1
Mean free path: ` = (v vrms
= √ 12
)
t π d2 n
rel rms
V
2 π d nV