Foundation Physics 1 (6E3Z1009) Lecture 1 PDF

Foundation Physics 1 (6E3Z1009) Dr Rasool Erfani Department of Engineering 6E3Z1009 Foundation Physics 1  Essentials 1EXAM20...

Foundation Physics 1 (6E3Z1009) Dr Rasool Erfani Department of Engineering 6E3Z1009 Foundation Physics 1  Essentials 1EXAM20 Multiple choice exam 30 Minutes Assessments Exam 2EXAM80 Standard questions 120 Minutes Department of Engineering 6E3Z1009 Foundation Physics 1  Lecture 1 Maths for Engineering & Physics Department of Engineering 6E3Z1009 Foundation Physics 1  Maths for Engineering and Physics Decimal Places (d.p.) and Significant Figures (s.f.) Indices Lecture 1 - Base units Outline Derived units Prefixes Transposition of formulae Department of Engineering 6E3Z1009 Foundation Physics 1  Maths is the cornerstone of Physics and the bedrock of Engineering Department of Engineering 6E3Z1009 Foundation Physics 1  Decimal places (d.p.) Take the number 17925.2054 3 d.p. 17925.205 2 d.p. 17925.21 1 d.p. 17925.2 Department of Engineering 6E3Z1009 Foundation Physics 1  Significant Figures (s.f.) 5|3879 to 1 significant figure is 50,000 53,879 53|879 to 2 significant figures is 54,000 0.005|089 to 1 significant figure is 0.005 0.005089 0.0050|89 to 2 significant figures is 0.0051 Take the number 791.25 4 s.f. 791.3 2 s.f. 790 3 s.f. 791 1 s.f. 800 Department of Engineering 6E3Z1009 Foundation Physics 1  Indices or Powers 104 – base 10 index 4 25 – base 2 index 5 63 – base 6 index 3 x2 – base x index 2 5a – base 5 index a ex – base e index x 4 to the power of 3 A log is an index and index is a log ➔ Department of Engineering 6E3Z1009 Foundation Physics 1  Positive and Negative Indices Inverse and Reciprocals Number In terms of 10s In term of indices 1000 10x10x10 103 Dividing by 10 100 10x10 102 10 10 101 1 10 100 10 0.1 1 1 10-1 or 101 10 0.01 1 1 10-2 or 102 10 × 10 0.001 1 1 10-3 or 103 10 × 10 × 10 Department of Engineering 6E3Z1009 Foundation Physics 1  Worth learning 𝑎0 = 1 𝑎1 = 𝑎 1 −n n =a a 1 over 𝑎𝑛 (or inverse or reciprocal of 𝑎𝑛 ) equals a to the power of minus n Department of Engineering 6E3Z1009 Foundation Physics 1  The Laws of Indices 103 x 102 = (10 x 10 x 10) x (10 x 10) = 10 x 10 x 10 x 10 x 10 = 105 = 10(3+2) 105 10 × 10 × 10 × 10 × 10 2 = = 10 × 10 × 10 10 10 × 10 3 = 10 = 105−2 (102 )3 = 102 × 102 × 102 = (10 × 10) × (10 × 10) × (10 × 10) = 106 = 102×3 Department of Engineering 6E3Z1009 Foundation Physics 1  Fractional and Decimal Indices (1) 𝟏 𝟏 𝟏 𝟏 Let’s explore ( + ) 𝒂𝟐 × 𝒂𝟐 = 𝒂𝟐 𝟐 = 𝒂𝟏 = 𝒂 Q. What two identical numbers multiplied together to give a? A. The square root of a. 𝟏 𝟐 So 𝒂 𝟐 is the same as 𝒂 Similarly 𝒂𝟎.𝟓 × 𝒂𝟎.𝟓 = 𝒂(𝟎.𝟓+𝟎.𝟓) = 𝒂𝟏 = 𝒂 So 𝒂𝟎.𝟓 is also 𝟐 𝒂 Department of Engineering 6E3Z1009 Foundation Physics 1  Fractional and Decimal Indices (2) Exploring further 1 1 1 1 1 1 (3 + 3 + 3) 𝑎3 × 𝑎3 × 𝑎3 =𝑎 = 𝑎1 = 𝑎 Q. What three identical numbers multiplied together give a? A. The cubed root of a. 1 3 So: 𝑎 = 3 𝑎 0.3ሶ 3 Similarly you can show that 𝑎 is also 𝑎 Department of Engineering 6E3Z1009 Foundation Physics 1  Fractional and Decimal Indices (3) 𝟏 − Beware of negative indices like 𝒂 𝟐 The negative index means the result is an inverse i.e. 1 − 1 𝑎 2 = 1 𝑎2 So in general So when the base a is a positive number the result is never 1 negative for example a n = a n 1 − 1 1 1 E.g. 100 2 = 1 = = = 0.1 100 10 1002 What about when the base a is a negative number? Department of Engineering 6E3Z1009 Foundation Physics 1  Units and prefixes Why are they so important in science and engineering? Department of Engineering 6E3Z1009 Foundation Physics 1  CAD for Porsche 550 Units are in thou (0.001inch) Units are in mm Department of Engineering 6E3Z1009 Foundation Physics 1  Units Industrial mixer tank – they have optimum effective with blades at certain height above base. Did you spot the change in unit? What would happened to the position of the blades and the effectiveness of the mixing tank if you didn’t and you thought it was 13 inches? Department of Engineering 6E3Z1009 Foundation Physics 1  Physical Quantities and SI units There are 7 physical properties from which all other physical properties can be related using equations. These 7 physical quantities each have their own SI unit and abbreviation. Department of Engineering 6E3Z1009 Foundation Physics 1  Physical Quantities and SI units From these 7 base units the units of all other physical properties are derived. Physical Equation Derived unit quantity Using equation Usual form Name velocity 𝐝𝐢𝐬𝐩𝐥𝐚𝐜𝐞𝐦𝐞𝐧𝐭 𝐦 m s-1 - 𝐭𝐢𝐦𝐞 𝐬 acceleration 𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲 m s−1 m s-2 - 𝐭𝐢𝐦𝐞 s force mass * acceleration kg * m s-2 kg m s-2 N (Newton) kg m2 s-2 work force * distance kg m s-2 ∗m N m (Joule) power 𝐰𝐨𝐫𝐤 kg m2 s−2 kg m2 s-3 W( Watt) 𝐭𝐢𝐦𝐞 𝐬 pressure 𝐟𝐨𝐫𝐜𝐞 kg m s−2 kg m-1 s-2 Pa (Pascal) Department of Engineering 𝐚𝐫𝐞𝐚 6E3Z1009 Foundation Physics 1 𝐦𝟐  Prefixes Prefixes and their abbreviations are a short hand way of expressing big or small quantities using powers of 10 2,740,000 m = 2.74 x 106 m = 2.74 megametres or Mm 0.0017 m = 1.7 x 10-3 m = 1.7 millimetre or mm Department of Engineering 6E3Z1009 Foundation Physics 1  Scientific Notation Vs. Engineering Notation 24,530,000 m can be expressed in scientific or engineering notation. Scientific Notation: Expressing numbers using any power of ten as long as the coefficient (the number before the power of 10) is between 1 and 10. 2.453 x 107 m Engineering Notation: Expressing numbers using only the powers of 10 that are divisible by 3 and coefficients that are between 1 and 1000. 24.53 x 106 m Engineering notation allows easy selection of a prefix in this case megametres or Mm. Scientific or Engineering notation is sometimes written in the form 2.453E+7 m where the E stands for exponent. Department of Engineering 6E3Z1009 Foundation Physics 1  In science and engineering, a formula (or equation) relates or expresses an unknown physical property in terms of other physical properties: Transposing E = mc2 Formulae (1) Relating the energy of an object (E) to its mass (m) and the speed of light (c). E is the subject of the equation. Department of Engineering 6E3Z1009 Foundation Physics 1  Transposing Formulae (2) Sometimes other elements of the formula are unknown. To find the unknown values, you have to transpose/rearrange the original formula. Rearranging the physical properties to find a new formula that express the unknown quantity as the subject in terms of other properties. You may have to use mathematic operations like multiply, divide, add, subtract, square, square- root etc., to transpose different formula. To transpose a formula, do the opposite mathematical operations to those that are in the base equation. Department of Engineering 6E3Z1009 Foundation Physics 1  Example 1 𝑪 = 𝝅𝒅 Formula to find circumference of a circle from its diameter If d = 2 then 𝐂 = 𝝅*2 = 2𝝅 What if we know the circumference and want to find the diameter? We make d the subject of this formula as in d = something Step 1 Start with the formula 𝐶=𝜋𝑑 Step 2 Divide both sides of the formula by π 𝑪 𝝅𝒅 = 𝝅 𝝅 Step 3 Cancel out the π’s from RHS 𝑪 =𝒅 𝝅 Step 4 The new formula will let you work out the diameter 𝑪 from any given circumference 𝒅= Department of Engineering 6E3Z1009 Foundation Physics 1 𝝅  Example 2 Formula for velocity (v) of an object from its initial v = u + at velocity (u), acceleration (a), and time taken (t) If u = 2, a = 4 and t = 3 then v = 2 + 4*3 = 14 But what if we know v, u, and a , and want to find the time taken (t)? Make t the subject of this formula as in t = something Step 1 Start with the formula 𝑣=𝑢+𝑎𝑡 Step 2 Subtract u from both sides of the formula 𝑣 − 𝑢 = 𝑢 − 𝑢 + 𝑎𝑡 Step 3 Cancel out the u’s from RHS 𝑣 − 𝑢 = 𝑎𝑡 Step 4 Divide both sides by a 𝑣 − 𝑢 𝑎𝑡 = 𝑎 𝑎 Step 5 Cancel out the a’s from RHS 𝑣−𝑢 =𝑡 𝑎 Step 6 The new formula will let you work out the time t for any given u, v and a 𝑣−𝑢 𝑡= Department of Engineering 6E3Z1009 Foundation Physics 1 𝑎  Shott, Milo. (1989). Foundation mathematics for non-mathematicians Open University Press. [eBook – MMU library see Moodle Full Reading List] Further Reading Maths online resource: http://www.mathscentre.ac.uk/ Department of Engineering 6E3Z1009 Foundation Physics 1