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Transcript of PHYSICS FORMULAE AND DATA BOOKLET - WordPress.com · PHYSICS FORMULAE AND DATA BOOKLET This booklet...
THE UNIVERSITY OF MALTA MATSEC SUPPORT UNIT
PHYSICS FORMULAE AND
DATA BOOKLET
This booklet is not to be removed from the examination room or marked in any way.
THE UNIVERSITY OF MALTA MATSEC SUPPORT UNIT
PHYSICS FORMULAE AND
DATA BOOKLET The following equations and formulae may be useful in answering some of the questions in the examination.
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Table of Contents
Uniformly accelerated motion.......................................................................... 3
Mechanics ................................................................................................................. 3
Circular motion and rotational dynamics .................................................... 3
Simple harmonic motion .................................................................................... 4
Ray optics ................................................................................................................. 4
Current electricity ................................................................................................. 5
Alternating current ............................................................................................... 5
Stationary waves ................................................................................................... 6
Wave motion ........................................................................................................... 6
Fields .......................................................................................................................... 6
Capacitance .............................................................................................................. 7
Inductance ................................................................................................................ 7
Electromagnetism ................................................................................................. 8
Temperature ........................................................................................................... 8
First and second laws of thermodynamics .................................................. 9
Gases ........................................................................................................................... 9
Materials ................................................................................................................... 9
Heat transfer ........................................................................................................ 10
Quantum phenomena ....................................................................................... 10
Radioactivity ........................................................................................................ 10
Mathematical Formulae ................................................................................... 11
Physical Constants ............................................................................................. 12
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Uniformly accelerated motion
Equations of motion:
𝑣 = 𝑢 + 𝑎𝑡 𝑣2 = 𝑢2 + 2𝑎𝑠
𝑠 = (𝑢 + 𝑣
2) 𝑡
𝑠 = 𝑢𝑡 +1
2𝑎𝑡2
Mechanics
Newton’s second law: 𝐹 = 𝑚𝑎 =𝑑(𝑚𝑣)
𝑑𝑡
Kinetic Energy: 𝐾𝐸 =1
2𝑚𝑣2
Potential Energy: 𝑃𝐸 = Δ(𝑚𝑔ℎ)
Mechanical Work Done: 𝑊 = 𝐹Δ𝑑
Power: 𝑃 = 𝐹𝑣
Momentum: 𝑝 = 𝑚𝑣
Circular motion and rotational dynamics
Angular speed: 𝜔 =𝑑𝜃
𝑑𝑡=
𝑣
𝑟
Angular acceleration: 𝛼 =𝑑𝜔
𝑑𝑡=
𝑎
𝑟
Centripetal acceleration: 𝑎 =𝑣2
𝑟
4
Centripetal force: 𝐹 =𝑚𝑣2
𝑟
Period: 𝑇 = 2𝜋𝑟
𝑣
Torque: 𝜏 = 𝐼𝛼
Work done in rotation: 𝜏𝜃 = Δ (1
2𝐼𝜔2)
Simple harmonic motion
Displacement: 𝑥 = 𝑥0 sin(𝜔𝑡 + 𝜙)
Velocity: 𝑣 = 𝜔𝑥0 cos(𝜔𝑡 + 𝜙)
𝑣 = ±𝜔√𝑥02 − 𝑥2
Acceleration: 𝑎 = −𝑘𝑥 = −𝜔2𝑥
Period: 𝑇 =1
𝑓=
2𝜋
√𝑘=
2𝜋
𝜔
Mass on a light spring: 𝑇 = 2𝜋√𝑚
𝑘
Ray optics
Refractive index: 𝑛1 sin 𝜃1 = 𝑛2 sin 𝜃2
𝑛1
2 =sin 𝜃1
sin 𝜃2=
𝑣1
𝑣2
𝑛1
3 = 𝑛1
2 × 𝑛2
3
Thin lenses: 1
𝑓=
1
𝑢+
1
𝑣
(real is positive)
5
1
𝑓=
1
𝑣−
1
𝑢 (Cartesian)
Magnification: 𝑚 =
𝑣
𝑢=
ℎ𝑖
ℎ𝑜
(real is positive)
𝑚 = −𝑣
𝑢= −
ℎ𝑖
ℎ𝑜 (Cartesian)
Current electricity
Ohm’s Law: 𝑉 = 𝐼𝑅
Current: 𝐼 = 𝑛𝐴𝑣𝑒
Resistors in series: 𝑅𝑇𝑂𝑇𝐴𝐿 = 𝑅1 + 𝑅2 + ⋯
Resistors in parallel: 1
𝑅𝑇𝑂𝑇𝐴𝐿=
1
𝑅1+
1
𝑅2+ ⋯
Power: 𝑃 = 𝐼𝑉 = 𝐼2𝑅 =𝑉2
𝑅
Resistivity: 𝜌 =𝑅𝐴
𝑙
Temperature coefficient: 𝛼 =𝑅𝜃 − 𝑅0
𝑅0𝜃
Alternating current
For sinusoidal alternating current:
𝐼 = 𝐼0 sin 2𝜋𝑓𝑡
Root mean square for sinusoidal alternating current and voltage:
𝐼𝑟𝑚𝑠 =𝐼0
√2 𝑉𝑟𝑚𝑠 =
𝑉0
√2
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Reactance: 𝑋𝐿 = 2𝜋𝑓𝐿 𝑋𝐶 =1
2𝜋𝑓𝐶
Stationary waves
Speed of waves on strings: 𝑣 = √𝑇
𝜇
Wave motion
Velocity of a wave: 𝑣 = 𝑓𝜆
Two slit interference: 𝑠 =𝜆𝐷
𝑑
Diffraction grating: 𝑑 sin 𝜃 = 𝑛𝜆
Single slit diffraction: 𝜃 =𝜆
𝑎
Diffraction of circular aperture: sin 𝜃 ≈ 𝜃 = 1.22
𝜆
𝑎
Fields
Electric field strength: 𝐸 = 𝐹
+𝑞= −
𝑑𝑉
𝑑𝑟
Uniform field: 𝐸 =𝐹
+𝑞=
𝑉
𝑑
Force between point charges:
𝐹 = 𝑄1𝑄2
4𝜋𝜀𝑜𝑟2
Electric field strength of a point charge:
𝐸 = 𝑄
4𝜋𝜀𝑜𝑟2
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Force between point masses:
𝐹 = 𝐺𝑀1𝑀2
𝑟2
Electric potential: 𝑉 = 𝑄
4𝜋𝜀0𝑟
Gravitational potential: 𝑉𝐺 = 𝐺𝑀
𝑟
Work: 𝑊 = 𝑄𝑉 = Δ (1
2𝑚𝑣2)
Capacitance
Charge on a capacitor: 𝑄 = 𝐶𝑉
Capacitance of parallel plates:
𝐶 =𝜀0𝜀𝑟𝐴
𝑑
Capacitors in parallel: 𝐶𝑇𝑂𝑇𝐴𝐿 = 𝐶1 + 𝐶2 + ⋯
Capacitors in series: 1
𝐶𝑇𝑂𝑇𝐴𝐿=
1
𝐶1+
1
𝐶2+ ⋯
Energy stored: 𝑊 =1
2𝐶𝑉2
Charging: 𝑄 = 𝑄0(1 − 𝑒−𝑡
𝑅𝐶⁄ )
Discharging: 𝑄 = 𝑄0𝑒−𝑡
𝑅𝐶⁄
Inductance
Mutual inductance: 𝑀 = −𝐸
𝑑𝐼𝑑𝑡⁄
8
Self-inductance: 𝐿 = −𝐸
𝑑𝐼𝑑𝑡⁄
Energy stored: 𝑊 =1
2𝐿𝐼2
Electromagnetism
Force on wire: 𝐹 = 𝐵𝐼𝑙
Torque on a rectangular coil:
𝜏 = 𝐵𝐴𝑁𝐼
Force on moving charge: 𝐹 = 𝐵𝑄𝑣
Magnetic flux: Φ = 𝐵𝐴
Field inside a solenoid: 𝐵 = 𝜇0𝜇𝑟𝑛𝐼
Field near a long straight wire:
𝐵 = 𝜇0
𝐼
2𝜋𝑟
Induced emf: 𝐸 = −𝑁𝑑
𝑑𝑡
Emf induced in a moving conductor:
𝐸 = 𝐵𝑙𝑣
Simple alternator emf: 𝑉 = 𝑉0sin (𝜔𝑡 + 𝜙)
Hall voltage: 𝑉𝐻 =𝐵𝐼
𝑛𝑄𝑡
Temperature
Temperature (K): 𝑇 = 273.16𝑃
𝑃𝑡𝑟K
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Celsius scale: 𝜃(°C) = 𝑇(K) − 273.15 K
First and second laws of thermodynamics
First law of thermodynamics:
Δ𝑈 = Δ𝑄 + Δ𝑊
Ideal heat engine: η = 1 −𝑇𝑐
𝑇ℎ
Gases
Ideal gas equation: 𝑃𝑉 = 𝑛𝑅𝑇
Kinetic theory of an ideal gas:
𝑃𝑉 =1
3𝑁𝑚⟨𝑐2⟩
Boltzmann’s constant: 𝑘 =𝑅
𝑁A
Principal molar heat capacities of an ideal gas:
𝛾 =𝐶𝑃
𝐶𝑉 𝐶𝑃 − 𝐶𝑉 = 𝑅
Adiabatic process: 𝑃𝑉𝛾 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Materials
Hooke’s law: 𝐹 = 𝑘Δ𝑥
Stress: 𝜎 =𝐹
𝐴
Strain: 𝜀 =Δ𝑙
𝑙
Young’s modulus: 𝑌 =𝜎
𝜀
10
Energy stored in a stretched wire: 𝐸 =
1
2𝑘(∆𝑙)2
Heat transfer
Thermal conduction: 𝑑𝑄
𝑑𝑡= −𝑘𝐴
𝑑𝜃
𝑑𝑥
Quantum phenomena
Quantum energy: 𝐸 = ℎ𝑓
Mass-energy: 𝐸 = 𝑚𝑐2
Photoelectric effect: ℎ𝑓 = 𝜙 + (1
2𝑚𝑣2)
𝑚𝑎𝑥
Energy levels: Δ𝐸 = 𝐸2 − 𝐸1 = ℎ𝑓 =ℎ𝑐
𝜆
De Broglie wavelength: 𝜆 =ℎ
𝑚𝑣
Radioactivity
Decay rate: 𝑑𝑁
𝑑𝑡= −𝜆𝑁
𝐴 = 𝜆𝑁
𝑁 = 𝑁0 𝑒−𝜆𝑡
Half-life: 𝑇1
2=
ln 2
𝜆
Absorption law for gamma radiation:
𝐼 = 𝐼0𝑒−𝜇𝑑
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Mathematical Formulae
Surface area of a sphere: 𝑆 = 4𝜋𝑟2
Volume of a sphere: 𝑉 =4
3𝜋𝑟3
Surface area of a cylinder: 𝑆 = 2𝜋𝑟ℎ + 2𝜋𝑟2
Volume of a cylinder: 𝑉 = 𝜋𝑟2ℎ
Logarithms: ln(𝑥𝑛) = 𝑛ln𝑥
ln(𝑒𝑘𝑥) = 𝑘𝑥
Equation of a straight line: 𝑦 = 𝑚𝑥 + 𝑐
Relationship between cosine and sine:
sin(90° − 𝜃) = cos 𝜃
Small angles: sin 𝜃 ≈ tan 𝜃 ≈ 𝜃(𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠)
12
Physical Constants
Acceleration of free fall on and near the Earth’s surface
𝑔 = 9.81 m s−2
Gravitational field strength on and near the Earth’s surface
𝑔 = 9.81 N kg−1
Mass of the Earth 𝑀𝐸 = 6.00 × 1024 kg
Radius of the Earth 𝑟𝐸 = 6.37 × 106 m
Boltzmann constant 𝑘 = 1.38 × 10−23 J K−1
Molar gas constant 𝑅 = 8.31 J K−1 mol−1
Avogadro’s constant 𝑁A = 6.02 × 1023 mol−1
Coulomb’s law constant 𝑘 =1
4𝜋𝜀0= 8.99 × 109 N m2 C−2
Charge of an electron 𝑒 = −1.60 × 10−19 C
Rest mass of an electron 𝑚e = 9.11 × 10−31 kg
Rest mass of a proton 𝑚p = 1.67(3) × 10−27 kg
Rest mass of a neutron 𝑚n = 1.67(5) × 10−27 kg
Electronvolt 1 eV = 1.60 × 10−19 J
Gravitational constant 𝐺 = 6.67 × 10−11 N m2 kg−2
Permittivity of free space 𝜀0 = 8.85 × 10−12 F m−1
Permeability of free space 𝜇0 = 4𝜋 × 10−7 H m−1
Planck constant ℎ = 6.63 × 10−34 J s
Speed of light in a vacuum 𝑐 = 3.00 × 108 m s−1
13
Unified atomic mass unit 1 u = 1.66 × 10−27 kg