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PHYSICS FORMULAE
AND
DATA BOOKLET
This booklet is not to be removed from the examination room or
marked in any way.
L-UNIVERSITA` TA’ 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.
2
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 ............................................................................................. 11
3
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
6
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
7
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
9
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
11
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)
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
12
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
Unified atomic mass unit 1 u = 1.66 × 10−27 kg