physics formulae and data booklet - … ta’ malta matsec support unit physics formulae and data...

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

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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)

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