PHYS1121-1131-1141 Formula sheet and data sheet PDF

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Formula Sheet - PHYS1121/1131/1141 Rotational Dynamics

Kinematics Constant acceleration:

τ = rF sin θ = Iα

1 x = x0 + vx0 t + axt2 2 v = vx0 + axt vx2 = v2x0 + 2ax(x − x0 )

I=

Circular motion: v = rω

v2 r

ac =

dθ dt

ω=

mi ri2

or

~L = ~r × ~p Z

X

Hooke’s law:

F~ = m~a

f = −kx

I=

Z

r2 dm

~τ = ~r × ~F

τ dθ

K=

L = mrv sin θ

Newton’s 2nd law: X d~p F~ = dt

dω dt

=

i

W =

Dynamics

X

α=

2 M R2 5

or

1 2 Iω 2

= Iω

solid sphere

Thermal Physics

Work and Power: ∆L = αL∆T

dW = F~ · d~x dW = F v cos θ P = dt

Q = mc∆T

1 mv2 2

F =−

U = mgh

U=

dU dx

1 2 kx 2

1 3 Kavg = kB T = mv2rms 2 2 ∆Eint =

Momentum: ~p = m~v Centre of mass: P mi~ri ~rcm = M

=

or

R

~rdm M

Gm1 m2 r2

U =−

Kepler’s 2nd Law:

Kepler’s 3rd Law: 

4π 2 GMS



for monatomic gas:

γ=

a3

CP CV

P = σAeT 4 ∆Eint = Q + W CP − CV = R

f =3

for diatomic gas: 0 < T . 100 K :

dA L = 2MP dt

T2 =

Gm1 m2 r

f f N kB ∆T = nR∆T 2 2

P V γ = constant    dT  P = kA   dx

dW = −P dV 1 CV = f R 2

Gravitation |F | =

Q = mL

P V = nRT = N kB T

Energy: K=

∆V = βV ∆T

f =3

100 . T . 1000 K :

f =5

T & 1000 K :

f =7

Trigonometric Identities

Waves and Oscillations T µ

v=

s

ω = 2πf

k=

2π λ

v=

s

F = −kx T = 2π

s

B ρ

v = fλ 1 f

T =

∆Pmax = vρωSmax k ω = m

l g

2

  I β = 10 log10 I0 power I= area 1 2 ρvω 2 Smax 2   c ± vo ′ f =f c ∓ vs I=

f=

1 2π

r

k m

 A∓B cos sin A ± sin B = 2 sin 2     A−B A+B cos cos A + cos B = 2 cos 2 2     A−B A+B cos A − cos B = −2 sin sin 2 2 

s

sin2 θ + cos2 θ = 1 a2 = b2 + c2 − 2bc cos A

Standard Integral

fbeat = f1 − f2

T µ

λn =

λ n

Greek Letters Capitals, when shown, appear second. Lower case phi can be displayed in 2 ways, either is acceptable. To enter the capital version of a Greek letter in online quizzes, capitalise the first letter of the name (e.g. Gamma).

Symbol

Name

Symbol

Name

α β γ, Γ δ, ∆ ǫ ζ η θ, Θ ι κ λ, Λ

alpha beta gamma delta epsilon zeta eta theta iota kappa lambda

µ ν ξ, Ξ π, Π ρ σ, Σ υ, Υ φ, ϕ, Φ χ ψ, Ψ ω, Ω

mu nu xi pi rho sigma upsilon phi chi psi omega



cos (A ± B) = cos A cos B ∓ sin A sin B

y(x, t) = A sin (kx − ωt + φ) n fn = 2L



sin (A ± B) = sin A cos B ± cos A sin B

I0 = 10−12 W · m−2 P I= 4πr2 1 P = µvω 2 A2 2

A±B 2

Z

1 dx = ln(x) + C x

Data Sheet Fundamental Physical Constants Quantity

Symbol

Avogadro’s constant Gas constant

N R

Elementary Charge Mass of electron

e me

Mass of neutron

mn

Mass of proton

mp

Boltzmann’s constant Speed of light Universal gravitation constant Stefan-Boltzmann Constant Wien law Constant Planck constant Permittivity of free space Permeability of free space Coulomb’s constant

kB c G σ b h ε0 µ0 ke =

Value

Unit 23

1 4πε0

6.022 × 10 8.314 0.08206 1.602 × 10−19 9.109 × 10−31 5.486 × 10−4 1.675 × 10−27 1.009 1.673 × 10−27 1.007 1.381 × 10−23 2.998 × 108 6.674 × 10−11 5.670 × 10−8 2.898 × 10−3 6.626 × 10−34 8.854 × 10−12 4π × 10−7 8.988 × 109

Thermal Data Quantity

particles/mol J · K−1 · mol−1 L · atm · K−1 · mol−1 C kg u kg u kg u J · K−1 m · s−1 N · m2 · kg−2 J · s−1 · m−2 · K−4 m·K J·s F · m−1 N · A−2 N · m2 · C−2

Value

Density of water at 25 C, ρw Volume of 1 mole ideal gas at 101.3 kPa(1 atm) and at 0◦ C (273 K) at 25◦ C (298 K) Specific Heat of Water Latent heat of vaporisation of water at constant pressure Latent heat of fusion of ice, Lf ◦

Other Data Quantity

Unit Conversions Energy Mechanical equivalent of heat Atomic mass unit One atmosphere (Standard air pressure)

1.000 × 10

2.241 × 10−2 2.447 × 10−2 4186 2.257 × 106 3.336 × 105

Value

Mass of Earth Average radius of Earth Mass of Moon Average Earth-Moon distance Mass of Sun Radius of Sun Average Earth-Sun distance Earth’s gravitational acceleration Reference Intensity near threshold of hearing

1 ev 1 cal 1u 1 atm

kg · m−3 m3 m3 J · kg−1 ·◦ C−1 J · kg−1 J · kg−1

Unit 24

g I0

Unit 3

5.972 × 10 6.371 × 106 7.346 × 1022 3.844 × 108 1.988 × 1030 6.964 × 108 1.496 × 1011 9.80 1.000 × 10−12

kg m kg m kg m m m · s−2 W · m−2

= 1.602 × 10−19 J = 4.186 J = 1.661 × 10−27 kg = 931.5 MeV · c−2 = 1.013 × 105 Pa

Thermal Properties of Metals You should use these values when answering questions in this course. Substance

Specific Heat c ( Jkg−1 K−1 )

Thermal conductivity k ( Wm−1 K−1 )

Linear expansion coefficient α Density at 0 ◦ C ( ◦ C−1 ) ( gcm−3 )

Aluminium Brass Copper Lead Steel

910 377 390 130 456

205.0 109.0 385.0 34.7 50.2

24 × 10−6 19 × 10−6 17 × 10−6 29 × 10−6 11 × 10−6

Thermal Properties of Water Some other properties of water can be found on the data sheet under thermal data. Quantity

Value

Unit

Melting Point (at 1 atm) Boiling Point (at 1atm) Volume expansion coefficient* β

0.000 100.0 207 × 10−6

◦

C C (◦ C)−1

◦

*at 20 ◦ C: you may assume it is constant between 15 ◦ C and 100 ◦ C

2.70 8.40 8.96 11.3 7.85...