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LF Mechanics Week1 – Vectors and SI Units Lecture 1a N Drury [email protected] Introduction and expectations

• These lectures will form a small part of your learning

• You should have paper, something to write with and a calculator alongside you as a minimum

• Take notes as you watch. Pause, rewind, rewatch.

• Add to your notes with further reading – this is not just a recommendation, it is essential!

• Practice, practice, practice! Assessment

• Formative – weekly problems

• Summative – Canvas problems, Assignments and Exam Learning Outcomes

• Explain Scalar and Vector quantities • Be able to apply the rules for mathematical operations on vectors • Describe the SI system of units, derived units and perform dimensional analysis to calculate units • Use prefixes with confidence • Understand what an appropriate number of significant figures is Scalars are quantities that only have magnitude (size).

Volume is a scalar quantity

e.g. a can holds 300ml of coke Scalar quantities

Scalar: quantity in physics that can be completely specified by its magnitude (size) and has no direction e.g. mass, volume, time, distance.

Scalar quantities can be added together as numbers (or subtracted), as long as they have the same units. mass A = 5kg, mass B = 15 kg total mass = mass A + mass B = 5+15 = 20 kg

Scalar quantities can be multiplied together as numbers (or divided); their units then do the same. mass = 10 kg, volume = 5 m3 density = mass/volume = 10/5 = 2 kg m-3 Vector quantities

Vector: quantity in physics that has both magnitude (size) and direction e.g. force, displacement, velocity, acceleration, torque, electric field, magnetic field. We represent vectors with arrows.

Notations used in formulas: 퐹Ԧ or F or 퐹ഥ for force, 푎Ԧ or a or 푎ത for acceleration, e.g. 퐹Ԧ = 푚푎Ԧ or F=ma Note that the equation above means that: • The size of the force vector is equal to the size of the acceleration vector multiplied by a number that corresponds to the mass and • The direction of the force vector is the same as the direction of the acceleration vector.

For the magnitude (size) of a vector we use: 푎Ԧ or 풂 or 푎ത or just a Properties of vectors

• Two vectors are equal when they have the same size and direction. They do not necessarily need to start from the same point e.g. a = b

a b

• When a vector is multiplied with a scalar its size is multiplied and its direction stays the same

a 3a http://physics.info/ve ctor-multiplication/

• The multiplication of one vector by another is not defined uniquely : o It can give a scalar quantity (dot product): a . b = ab cos θ where θ is the angle between the vectors when they start from the same point o It can give a vector quantity (cross product): a x b = ab sin θ n̂ where n̂ is a unit vector perpendicular to the plane formed by the two vectors Unit vector

• A unit vector is any vector having a magnitude of one and pointing in a specific direction. Trigonometry Adding vectors

• Vectors can be added only if they represent similar concepts e.g. you cannot add a force to a velocity vector • if vectors are at 90° to each other, they can be added using the parallelogram law: 푦 The components 4i and 3j are called Cartesian b=3j components of the vector

휃 a=4i 푥

• We can then use the notation R = 4i + 3j or R = (4,3) where i and j are the unit vectors on the axes 푥 and 푦 respectively Length and direction of resultant vector

• To find the length of the resultant vector R you use the Pythagoras theorem. Example: a= 4i R = 42 + 32 = 5 b = 3j

• To find the direction of the resultant vector you use the trigonometric number tan 휃 of the angle between the resultant vector R and the 푥-axis

Example: 3 tan 휃 = = 0.75 ⇒ 휃 = 36.87° 4 Examples

Add the vectors below: 1. 12i and 5j Ans: 13, 22.6° 2. 1i and 1j Ans: 1.4, 45° 3. 6i and 9j Ans:10.8, 56.3° LF Mechanics Week1 – Vectors and SI Units Lecture 1b N Drury [email protected] Learning Outcomes

Add the vectors below: 1. 12i and 5j Ans: 13, 22.6° 2. 1i and 1j Ans: 1.4, 45° 3. 6i and 9j Ans:10.8, 56.3° Examples

Add the vectors below: 1. 12i and 5j Ans: 13, 22.6° 2. 1i and 1j Ans: 1.4, 45° 3. 6i and 9j Ans:10.8, 56.3° Adding vectors (same starting point)

The parallelogram law applies even if the vectors are not at 90°, as long as the vectors have the same starting point. You can only add two vectors at a time.

r = a + b Adding vectors (tip to tail)

Vectors can also be added tip-to-tail. In this case the resultant vector has its start at the start of the first vector and its end at the end of the last vector.

R = a + b + c

c Subtracting vectors

For every vector a we can define vector -a which is one that has the same size but opposite direction. Example: -a a

Subtraction is equivalent to addition of the opposite. Example: a-b = a+(-b)

a -b a therefore

b b Examples

Find the answers for the vector subtractions below:

1. 5j- 12i Ans: 13, 157.4° 2. 1j- 1i Ans: 1.4, 135° 3. 9j- 6i Ans:10.8, 123.7° Examples

Find the answers for the vector subtractions below:

1. 5j- 12i Ans: 13, 157.4° 2. 1j- 1i Ans: 1.4, 135° 3. 9j- 6i Ans:10.8, 123.7° Examples

Subtract the j vector from the i: 1. 12i and 5j Ans: 13, -22.6° 2. 1i and 1j Ans: 1.4, -45° 3. 6i and 9j Ans:10.8, -56.3° Examples

Subtract the j vector from the i: 1. 12i and 5j Ans: 13, -22.6° 2. 1i and 1j Ans: 1.4, -45° 3. 6i and 9j Ans:10.8, -56.3° What are the following quantities measured in?

• Time • Length • Mass • Current • Temperature • Amount of substance • Light intensity Time – the second (s)

• The second is used to measure time. As well as enabling us to tell the time of the day, accurate timekeeping is key to satellite navigation systems, underpins the functioning of the internet and facilitates timestamping for transactions in financial trading. • Ancient civilisations used sundials and obelisks to tell the time, which was not very precise, and was restricted in cloudy weather or at night. These methods of timekeeping were based on the daily rotation of the Earth around its own axis. However this period of rotation is not regular enough to serve as a definition for modern-day applications. • Atomic clocks, which keep time using transition energies in atoms, revolutionised timekeeping. NPL developed the first operational caesium- beam atomic clock in 1955. This clock was so accurate that it would only gain or lose one second in three hundred years. Modern atomic clocks can be as much as a million times more accurate than this, and underpin satellite technology, like GPS or the internet. Length (m)

• We measure distances by comparing objects or distances with standard lengths. Historically, we used pieces of metal or the wavelength of light from standard lamps as standard lengths. Since the speed of light in vacuum, c, is constant, in the SI we use the distance that light travels in a given time as our standard. • For long distances, such as from the Earth to the moon, we simply measure the time light takes to travel the distance using an accurate timer; for shorter distances (when the time is too short to measure accurately enough), we use laser-based systems, where we ‘count’ the number of wavelengths of the laser light that correspond to the length we want to measure. Mass – the kilogram (Kg)

• Accurately measuring the mass of an object is essential in many applications, from administering the optimum dose of a drug to correctly manufacturing materials with the desired properties. • For more than a hundred years we compared the gravitational force on an object with the gravitational force on a reference piece of metal known as a 'standard weight'. The standard weight was in turn compared with the International Prototype of the Kilogram (IPK), held in the International Bureau of Weights and Measures in France, which was the ultimate reference for all mass measurements. However, the value of the IPK may have changed since it was produced in 1884; contamination, cleaning or just time may have increased or decreased its mass. • Since the revision of the SI on 20 May 2019, we can now compare the gravitational force on an object with an electromagnetic force using a Kibble balance. This allows the kilogram to be defined in term of a fixed numerical value of the Planck constant, a constant which will not change over time. Current – the ampere (A)

• The ampere, or 'amp' for short, measures electric current, which is a flow of electrons along a wire or ions in an electrolyte, as in batteries. Electric current allows us to power electrical devices, like smartphones or laptops and can even be used to operate a bus or car. • The ampere has only been in use for as long as we have had access to electricity – a small proportion of the history of measurement. The ampere definition exploits the fact that electric current is made up of a flow of billions of identical charged particles called electrons. We can create a standard ampere by using special nano-scale electric circuits that control the flow of electrons. Temperature – the kelvin (K)

• Prior to May 2019, all temperatures were defined relative to the triple point of water, the temperature at which water can exist as a solid (ice), a liquid and a gas (water vapour). The temperature at which this condition occurs was defined to be 273.16 K exactly, and every temperature measurement was fundamentally a measure of how much hotter or colder something was than this standard temperature. Whilst convenient for ambient temperatures, this definition increased the uncertainty of measurements at very high and very low temperatures. • Since May 2019, the kelvin and degree Celsius have been defined by taking a fixed numerical value of the Boltzmann constant, k or kB. This change in definition acknowledges that temperature is fundamentally a measure of the average energy of molecular motion. Amount of substance – the mole (mol)

• The mole is a base unit of the International System of Units (SI). • A mole is the amount of substance containing as many elementary entities as there are atoms in exactly 0.012 kilogram (or 12 grams) of carbon-12, where the carbon-12 atoms are unbound, at rest and in their ground state. • The mole is used to describe a practical quantity of material and is the link between the microscopic and macroscopic worlds, used to scale phenomena from the atomic up to 'relevant' sizes. As a result of the definition, the mole contains a defined number of entities, usually atoms or molecules. This number is the Avogadro constant (NA). 23 -1 • The current value for NA is: 6.022 141 79(30)x10 mol • This number is a dauntingly large figure. • This number of sand grains would cover the United Kingdom to a depth of about 40 centimetres. • There are about this number of human cells on Earth. • It would take you twenty thousand million million years to count this number of coins (counting about one coin per second). Light intensity – the candela (cd)

• The current definition of the candela was made in 1979, in terms of the watt at only one wavelength of light. It is defined as: • The luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watts per steradian (a unit of solid angle). So what?

• The point of these units, which were agreed upon in 1960, is to have a common understanding of the fundamental units of measurement. All other units of measurement are derived from these. LF Mechanics Week1 – Vectors and SI Units Lecture 1c N Drury [email protected] Learning Outcomes

SI is constructed from seven base units, which are adequate to describe most of the measurements used in science, industry and commerce. All other units are derived from these. Quantity Unit name Symbol length metre m mass Kilogram kg time second s electric current ampere A thermodynamic kelvin K temperature amount of mole mol substance luminous intensity candela cd Derived Units In calculations, units are treated in the same way as numbers are. Examples: m2 m-1 = m m-3 = 1/m3 m3/ m2 = m3 m-2 = m Examples of SI derived units expressed in terms of base units Derived Quantity SI derived unit Name Symbol area square metre m2 volume cubic metre m3 speed, velocity metre per second m/s metre per second acceleration m/s2 squared kilogram per cubic density, mass density kg/m3 metre cubic metre per specific volume m3/kg kilogram ampere per square current density A/m2 metre magnetic field strength ampere per metre A/m concentration (of amount of mole per cubic metre mol/m3 substance) candela per square luminance cd/m2 metre Examples Equation to use Derived unit I = Q/t current = charge/time Coulomb: unit of charge C = s · A f=1/T frequency =1/period Hertz: unit of frequency Hz= s-1 P=E/t power = energy/time Watt : unit of power W= m2 · kg · s-3 V=E/Q voltage = energy/charge Volt: unit of voltage V= m2 · kg · s-3 · A-1 C=Q/V capacitance = charge/voltage Farad: unit of capacitance F= m-2 · kg-1 · s4 · A2 R=V/I resistance = voltage/current Ohm: unit of resistance Ω = m2 · kg · s-3 · A-2 V = dΦ/dt voltage = change in flux/time Weber: unit of flux Wb = m2 · kg · s-2 · A-1 Examples Equation to use Derived unit I = Q/t current = charge/time Coulomb: unit of charge C = s · A f=1/T frequency =1/period Hertz: unit of frequency Hz= s-1 P=E/t power = energy/time Watt : unit of power W= m2 · kg · s-3 V=E/Q voltage = energy/charge Volt: unit of voltage V= m2 · kg · s-3 · A-1 C=Q/V capacitance = charge/voltage Farad: unit of capacitance F= m-2 · kg-1 · s4 · A2 R=V/I resistance = voltage/current Ohm: unit of resistance Ω = m2 · kg · s-3 · A-2 V = dΦ/dt voltage = change in flux/time Weber: unit of flux Wb = m2 · kg · s-2 · A-1 Standard Form and Prefixes

Standard form: there is always one non-zero digit before the decimal point e.g. 4.3 x 105 instead of 43 x 104 prefix symbol multiplier power of 10 giga G 1 000 000 109 000 mega M 1 000 000 106 kilo k 1 000 103 hecto h 100 102 deca da 10 101 deci d 0.1 10-1 centi c 0.01 10-2 milli m 0.001 10-3 micro μ 0.000 001 10-6 nano n 0.000 000 10-9 001 WhenWhen youyou areare given given a a variable variable with with a prefixa prefix you you must must convert convert it into it itsinto numerical its numerical equivalent equivalentin standard inform SI unitsbefore before you use you it in use an itequation. in any equation. FOLLOW THIS! Always start by replacing the prefix symbol with its equivalent multiplier.

For example: 0.16 μA = 0.16 x 10-6 A

3 km = 3 x 103 m

10 ns = 10 x 10-9 s

DO NOT get tempted to follow this further (for example: 0.16 x 10-6 A = 1.6 x 10-7 A and also 10 x 10-9 s = 10-8 s) unless you are absolutely confident that you will do it correctly. It is always safer to stop at the first step (10 x 10-9 s) and type it like this into your calculator.

NOW TRY THIS!

1.4 kW = 1.4x103 W 10 μC = 10x10-6 C = 1.0x10-5 C Use standard form for the -2 -1 24 cm = 24x10 m = 2.4x10 m final answer only. Do not 6 8 340 MW = 340x10 W = 3.4x10 W use it while you are -12 -11 46 pF = 46x10 F = 4.6x10 F working out the answer. 0.03 mA = 0.03x10-3 A = 3.0x10-5 A 52 Gbytes = 52x109 bytes = 5.2x1010 bytes 43 kΩ = 43x103 Ω = 4.3x104 Ω 0.03 MN = 0.03x106 N = 3.0x104 N WhenWhen youyou areare given given a a variable variable with with a prefixa prefix you you must must convert convert it into it itsinto numerical its numerical equivalent equivalentin standard inform SI unitsbefore before you use you it in use an itequation. in any equation. FOLLOW THIS! Always start by replacing the prefix symbol with its equivalent multiplier.

For example: 0.16 μA = 0.16 x 10-6 A

3 km = 3 x 103 m

10 ns = 10 x 10-9 s

NOW TRY THIS!

• While performing experiments, we should note the of the number of significant figures of each reading, to maintain a consistent level of precision in our calculations. Significant figures

• In physics, numbers represent values that have uncertainty and this is indicated by the number of significant figures (sf) in an answer. All answers should be given to an appropriate number of sf. Unless the question tells you otherwise, the number of sf in your calculated answer should be the same as the smallest number of sf in any of the values used in the calculation. For example, if the least accurate data used in a calculation is given to 4 sf, then the answer should be given to 4 sf. • When there is a decimal point, all digits are significant, except leading (leftmost) zeroes: 2.00 (3 sf) 0.0020 (2 sf) 200.1 (4 sf) 200.040 (6 sf) • Numbers without a decimal should be treated as if all digits (including trailing zeroes) are significant: 375 (3 sf) 200 should be treated the same as 2.00×102 (3 sf) 40 should be treated as 4.0×101(2 sf) Conversion of Units

Find the speed of rotation of a town near the equator, in km/h and in m/s. The Earth circumference is 40 075 km. Working out speed Speed (km/h)= distance (km)/time(h) = 40 075/24 = 1669.79 = 1700 km/h (note number of significant figures) Converting km/h to m/s 1 km = 1000m ⇒ 1000m/1km = 1 24 h = 24 x 60 min = 24 x 60 x 60 sec = 86400 s ⇒ 24 h/86400s = 1

푘푚 1000푚 24ℎ 1669.79 ∗1000∗24 푚 푚 1669.79 = = 463.83 = 460 m s-1 ℎ 1푘푚 86400푠 86400 푠 푠 Questions on unit conversion

Convert Joules (J) to electron-Volts (eV) and vice versa if 1J = 6.241509x1018 eV

1. How many eV conrrespond to 3 000 J? 2. How many J correspond to 2x1019 eV?

Worked answer for question 1: 6.241509x1018 eV 3000J=3000J = 18.72x1021 eV 1J

Worked answer for question 2: 1J 2x1019 eV = 2x1019 eV = 3.2J 6.241509x1018 eV Questions on unit conversion

Convert Joules (J) to electron-Volts (eV) and vice versa if 1J = 6.241509x1018 eV

1. How many eV correspond to 3 000 J? 2. How many J correspond to 2x1019 eV?

Worked answer for question 1: 6.241509x1018 eV 3000J=3000J = 18.72x1021 eV 1J

Worked answer for question 2: 1J 2x1019 eV = 2x1019 eV = 3.2J 6.241509x1018 eV Physical models

Physical model: a simplified representation of a real complex situation. A series of assumptions are used to simplify the reality.

Common assumptions • Surfaces are smooth, therefore frictional forces do not exist • Strings, ropes, rods are light, therefore they have zero weight • Strings and ropes are inextensible therefore tension stays constant • The air resistance is too small therefore it can be ignored • An object can be modelled as a particle therefore any rotational motion is ignored Hidden mathematical meanings

sentence mathematical meaning charge per unit time charge divided by time the product of voltage and multiply voltage with current current the sum of powers of the add all powers of lenses lenses uniform velocity constant velocity Summary

Review • OpenStax College Physics: Chapter 1 (p13-26 – Chapter 1: The nature of science and Physics) and Chapter 2 (p.32 – 35 – Chapter 2: Kinematics)

Read ahead • OpenStax College Physics: Chapter 4: Dynamics – Forces and Newton’s Laws of motion