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Formula Sheet - Physics 111
x
y
A^ ~
x
A y
A^ ~
θ
2D Vectors:
A = A x + A y = (A x , A y ) ∣ ∣ ∣ A
∣ = A =
A
2 x +^ A
2 y
A x = A cos θ A y = A sin θ
tan θ =
A y
A x
Algebra & Trigonometry:
ax
2
−b ±
b 2 − 4 ac
2 a
2 sin
2 x + cos (2x) = 1 cos (x + y) = cos x cos y − sin x sin y
2 cos
2 x − cos (2x) = 1 sin (x + y) = sin x cos y + cos x sin y
sin
2 x + cos
2 x = 1 2 cos x cos y = cos (x − y) + cos (x + y)
sec
2 x − tan
2 x = 1 2 sin x sin y = cos (x − y) − cos (x + y)
csc
2 x − cot
2 x = 1 2 sin x cos y = sin (x − y) + cos (x + y)
Conversion factors:
femto- (f) = 10
− 15
pico- (p) = 10
− 12
nano- (n) = 10
− 9
micro- (μ) = 10
− 6
milli- (m) = 10
− 3
centi- (c) = 10
− 2
kilo- (k) = 10
3
mega- (M) = 10
6
giga- (G) = 10
9
tera- (T) = 10
12
1 yd=3 ft = 36 in 1 in=2.54 cm
1 mi=1609 m 1 lb=4.448 N
1 gal=3.78541 L 1 m
3 =1000 L
1 atm= 1. 013 × 10
5 Pa 1 cal=4.186 J
=760 mm Hg 1 Cal=1000 cal
Geometry
Circle: Perimeter = 2πr
Area = πr
2
Sphere: Surface Area = 4πr
2
Volume =
πr
3
Cylinder: Surface Area = 2πr
2
Volume = πr
2 h
Few Physical Constants & Quantities:
REarth = 6. 38 × 10
6 m g = 9. 80 m/s
2
MEarth = 5. 97 × 10
24 kg G = 6. 67 × 10
− 11 Nm
2 /kg
2
Msun = 1. 99 × 10
30 kg R = 8. 314 J/mol · K
rsun−Earth = 1. 496 × 10
11 m kB = 1. 381 × 10
− 23 J/K
STP : 1 atm, 273 K NA = 6. 022 × 10
23 mol
− 1
vsound , air = 343 m/s σ = 5. 67 × 10
− 8 W/m
2 K
4
ρwater = 1000 kg/m
3
General Kinematics
v avg =
∆ r
∆t
r 2 − r 1
t 2 − t 1
v = lim ∆ t → 0
∆ r
∆t
a avg =
∆ v
∆t
v 2 − v 1
t 2 − t 1
a = lim ∆ t → 0
∆ v
∆t
constant acceleration 1D motion Kinematics
x = x 0 + v 0 t +
at
2 v = v 0 + at
v
2 = v
2 0 + 2a∆x^ v^ =^
v + v 0
Constant acceleration Kinematics 2D:
x = x 0 + v 0 x t +
a x t
2 v x = v 0 x + a x t v
2 x =^ v
2 0 x + 2a x ∆x
y = y 0 + v 0 y t +
a y t
2 v y = v 0 y + a y t v
2 y =^ v
2 0 y + 2a y^ ∆y
v x =
v x + v 0 x
v y =
v y + v 0 y
Projectile motion ( ax = 0 and ay = − g )
x = x 0 + v x t v x = constant.
y = y 0 + v 0 y t −
gt
2 v y = v 0 y − gt v
2 y = v
2 0 y − 2 g∆y
v y =
v y + v 0 y
v
2 = v
2 x +^ v
2 y v
2 = v
2 0 −^2 g∆y
Relative motion
r a , c = r a , b + r b , c v a , c = v a , b + v b , c a a , c = a a , b + a b , c
Forces
F = m a F (^) AB = − F (^) BA Fspring = −kx
F (^) g (≡ W ) = m g fk = μkN fs ≤ μsN
Circular motion
ω =
∆θ
∆t
α =
∆ω
∆t
s = Rθ v = Rω atan = Rα
a = a tan + a r arad =
v
2
R
= ω
2 R a =
a
2 tan +^ a
2 r
Constant α :
ω = ω 0 + αt ω
2 = ω
2 0 + 2α∆θ
ω =
ω + ω 0
θ = θ 0 + ω 0 t +
αt
2
Uniform Circular Motion (α = 0) : T =
f
2 π
ω
2 πR
v
Work and Energy
W =F d cos θ ∆KE = Wtotal KE =
mv
2
p
2
2 m
P =
W
t
= F v cos θ ∆P E = −Wcons ∆E = Wnon−cons
E =P E + KE ∆E = ∆P E + ∆KE
P Es =
kx
2 P Egrav = mgy
Gravitation
F (^) g = −
GmM
r 2
ˆ r g =
GM
r 2
T
2
r^3
4 π
2
GM
vcirc orbit =
GM
r
Momentum, Impulse, System of particles
F (^) net =
∆ p
∆t
= mtot a cm p = m v J = ∆ p = F (^) net , avg∆t
r cm =
m i r i ∑ m i
v cm =
m i v i ∑ m i
a cm =
m i a i ∑ m i
p i = mtot v cm If F (^) net = 0 ⇒ ∆ p = 0 ⇒ p = p
′
Elastic Collision:
KEbefore = KE
′ after vA , x − vB , x = −
v
′ A , x − v
′ B , x
Rigid-body and Rotational motion
KE = KEtr + KErot KErot =
Iω
2 KEtr =
mv
2 cm
τ = rF sin θ = r⊥F = rF⊥
τ = I~α W = τ ∆θ
L = rp sin θ = r⊥p = rp⊥ L = Iω P = τ ω
τ =
∆ L
∆t
if
~τ = 0 ⇒ ∆ L = 0 ⇒ L = L
′
Icm for few objects about symmetry axis:
Isolid sphere =
mr
2 Isolid cylinder =
mr
2
Ihollow sphere =
mr
2 Ihollow cylinder = mr
2
Irod =
m`
2 Irectangle =
m
a
2
2
a b
Elasticity
E =
F/A
∆/ 0
B = −
∆P
∆V /V 0
G =
F||/A
x/h
Fluids
P =
F
A
P = P 0 + ρgh ρ =
m
V
FB = ρFVdisplg
A 1 v 1 = A 2 v 2 = const P + ρgy +
ρv
2 = const
Simple Harmonic Motion
x = A cos(ωt) v = −Aω sin(ωt) a = −Aω
2 cos(ωt)
ωspring =
k
m
T =
f
2 π
ω
Tpendulum = 2π
`
g
Mechanical Waves
v = λf ω = 2πf v =
F
m/`
f =
T
〈P 〉 =
ω
2 A
2
μF I =
P
4 πr 2
sound:
v =
γRT
M
γair = 1. 40 β = (10 dB) log 10
I
I 0
fbeat = |f a − f b | I 0 = 10
− 12 W/m
2
fobs =
vsnd ± vobs
vsnd ± vsrc
fsrc
Interference:
d 2 − d 1 = n
λ
n = 0, ± 2 , ± 4 ,... (constructive)
n = ± 1 , ± 3 , ± 5 ,... (destructive)
Standing wave normal modes:
f n = nf 0 =
nv
2 `
n = 1, 2 , 3 ,... (both ends either fixed or free)
nv
4 `
n = 1, 3 , 5 ,... (one end fixed)
Temperature and Heat
TK = TC + 273.15 K TC =
(TF − 32
◦ C) ∆= α 0 ∆T
Qrad
t
= σA
T
4 − T
4 s
) Q
cond
t
=kA
T H − T C
`
∆V = βV 0 ∆T
Q = mc∆T Q =nCmolar∆T Q = mL
Ideal Gas
pV = nRT = N kT vrms =
3 kT
m
3 RT
M
n =
N
N A
m
M
〈KE〉 =
kT =
mv
2 rms
k =
R
N A
= 1. 381 × 10
− 23 J/K U =
nRT (monoatomic)
Thermodynamics
∆U = Q − W W = p∆V W = |QH| − |QL|
eengine =
W
QH
COPrefrig =
QL
|W |
COPideal pump
TH
TH − TL
eideal = 1 −
TL
TH
COPpump =
QH
W
COPideal refrig
TL
TH − TL
∆S =
Q
T
ideal :
TL
TH
QL
QH
