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Vectors and SI Units.Pdf 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 • 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 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° 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.
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