PHYS1001 Formula Sheet 2020 PDF

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Semester 2 Examinations

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PHYS1001 data & formulae

The following equations and data might be useful: Acceleration due to gravity at earth’s surface:

g = 9.80 m s−2

Speed of light in a vacuum

c = 3.00 × 108 m s−1

Permittivity of free space Mass of one proton Mass of one electron Charge of an electron

ǫ0 = 8.85 × 10−12 F m−1 mp = 1.673 × 10−27 kg me = 9.11 × 10−31 kg e = 1.60 × 10−19 C

Boltzmann’s constant Planck’s constant

kB = 1.38 × 10−23 J K−1 h = 6.626 × 10−34 J s = 4.136 × 10−15 eV s ~=

h = 1.055 × 10−34 J s 2π = 6.582 × 10−16 eV s σ = 5.67 × 10−8 W m−2 K−4 R∞ = 13.606 eV R = 8.314 J K−1 mol−1

The Stefan-Boltzmann constant Rydberg The universal gas constant

cwater = 4184 J kg−1 K−1

Specific heat of liquid water

cice = 2050 J kg−1 K−1

Specific heat of ice Heat of fusion for water

Lf,water = 3.34 × 105 J kg−1

Heat of vapourisation for water Speed of sound in air

Lv,water = 2.26 × 106 J kg−1 = 343 m s−2

Density of water (20◦ C and 1 atm) Conversion factors:

= 1.00 × 103 kg m−3 1 eV = 1.60 × 10−19 J

Area of a Sphere:

0 ◦ C = 273.15 K 1 L = 10−3 m3 A = 4πr 2

Prefixes: f = 10−15 , p = 10−12, n = 10−9 , µ = 10−6 , m = 10−3 , k = 103 , M = 106 , G = 109 , T = 1012

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Semester 2 Examinations

PHYS1001 data & formulae

Heat & Thermodynamics equations Thermal expansion:

∆L = αLi ∆T

Heating/Cooling:

Q = mc∆T

Thermal motion in a gas:

Kave,trans = 21 mv 2th = 32 kT

Heat Transfer by radiation:

4 Pnet = Pabs − Pem = eσA(T env − T 4)

Heat Transfer by conduction:

H=

First Law and Work:

∆U = Q + W

Ideal Gas Law:

pV = nRT

Internal energy (ideal gas):

U = 32 nRT (monatomic)

γ (ideal gas):

γ=

∆V = β Vi ∆T Q = mL

c = kA Th −T L

Q t

β = 3α

vth =

p

3kT /m

H is heat flow in watts. RV W = − V f p dV i

Where n is the number of moles of gas. ∆U = nCV ∆T

Cp CV

γmonatomic = γdiatomic =

5 3

7 5

γpolyatomic =

4 3



Vf Vi



Wadiabatic =

Work (ideal gas):

Wisothermal = −nRT ln

Specific Heat (ideal gas):

Q = nCV ∆T

Q = nCp ∆T

Cp = CV + R

CV = f2 R

Adiabatic process in ideal gas:

pV γ = constant

Entropy change:

∆S =

Q T

T V γ−1 = constant R f dQ (constant T ) ∆S = i T

pf Vf −pi Vi γ−1

∆S = mc ln

Mechanics equations

p = mv v=ω×r

dp dt τ =r×F F=

L = Iω

τ=

L=r×p

Iz =

dL dt X

mi ri2

i

1 Ktrans = mv 2 2 dU F (x) = − dx

Krot

1 = Iω 2 2

f ≤ µs N

f = µk N

GMm Ugrav = − + U0 r Z x F (x)dx = U (x0 ) − U (x) x0

dr Circular motion: v = = r ϑ˙ eˆϑ dt Kinematics:

Fgrav

GMm =− 2 r

v = v0 + at

dv a= = −rϑ˙2 ˆer = −ω 2 r dt 1 x = x0 + v0 t + at2 v 2 = v 20 + 2a(x − x0 ) 2

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

Tf Ti

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Semester 2 Examinations

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PHYS1001 data & formulae

Waves & Optics equations 1 T

Frequency:

f=

Angular Frequency:

ω = 2πf

Wave number:

k=

Average Power of Wave on a String:

P¯

Intensity:

I=

Standing Waves:

L=

Doppler Effect:

2π λ = 21 µvω 2 A2 P A

n 2 λ, L=m 4 λ, f ′ = 1±f u v

n = 1, 2, 3, ... m = 1, 3, 5, ....

Snell’s Law:

n1 sin θ1 = n2 sin θ2

Refractive Index:

n=

Critical Angle:

sin θc =

Double Slit (Bright Fringes)

d sin θ = mλ,

Double Slit (Dark Fringes)

d sin θ = (m +

N Slit (Bright Fringes)

d sin θ = mλ,

N Slit (Dark Fringes)

d sin θ =

Single Narrow Slit (Destructive):

a sin θ = mλ,

Rayleigh Criterion (single slit):

θmin =

Law of Malus:

I = I0 cos2 θ

Trigonometric Identities:

sin( π2 ± x) = cos x

c v n2 n1

m N λ,

m = 0, 1, 2, ... 1 2 )λ,

m = 0, 1, 2, ...

m = 0, 1, 2, ... m = 1, 2, 3, ... AN D m 6= multiple of N m = 1, 2, 3, ...

λ a

sin α ± sin β = 2 sin 12 (α ± β) cos 21(α ∓ β) cos α + cos β = 2 cos 12 (α + β) cos 12 (α − β) cos α − cos β = 2 sin 12 (α + β) sin 12 (β − α)

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Semester 2 Examinations

PHYS1001 data & formulae

Electricity equations Vsphere = 43 πr 3

Asphere = 4πr2

Acircle = πr 2

Ccircle = 2πr

Electric Force:

~ = qE ~ F

Electrical Potential:

V =

Potential difference:

∆V = −

Field of point charge:

~ = E

Potential of point charge:

V =

Gauss’s Law:

ΦE =

Electric Field from Potential:

~E = −∇V ~ = −( ∂V , ∂V , ∂V ) ∂x ∂y ∂z

Field of Sphere:

~E =

U q

Rb

~ · d~l E

a 1 q ˆ 4πǫ0 r2 r 1 q 4πǫ0 r

~ · dA ~ = E

H

Field near conducting sheet:

1 q 4πǫ0 r2 Esheet = σǫ0

Field near non-conducting sheet:

Esheet =

Field of linear charge distribution:

Eline =

Capacitance:

C=

Parallel plate Capacitor:

Ck =

qenclosed ǫ0

rˆ

σ 2ǫ0

λ 2πǫ0 r

Q V ǫ0 A d

∆V = Ed UC = 12 CV 2

Capacitor Energy:

Breakdown of Classical Physics equations E c = fλ, f= , h p 2 2 E = |~ p | c + m2 c 4 ,

E = |~ p|c

λ=

h p

I(T ) = σT 4 λmax T = 2.898 × 10−3 m K λ sin θ = m , m = 1, 2, 3, . . . a h L = n~ = n , n = 1, 2, 3, . . . 2π ~ ~ ∆p ∆x ≥ , ∆E ∆t ≥ 2 2 2 RZ E = − 2 , where R = 13.606 eV n

for photons

Kmax = hf − Φ, h (1 − cos φ) mc p 2m(U − E) −2bL T ≈e , where b = ~ n2 ǫ0 h2 rn = , me vr = n~ Zπe2 me

∆λ = λ′ − λ =

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Semester 2 Examinations

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PHYS1001 data & formulae

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Semester 2 Examinations

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PHYS1001 data & formulae

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