01 Measurement (H2) 2020 - Student

1-1 ST. ANDREW’S JUNIOR COLLEGE JC 1 PHYSICS 2020 TOPIC 1: MEASUREMENT Learning Outcomes: Candidates should be able to: a. recall the following base quantities and their SI units: mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol). b. express derived units as products or quotients of the base units and use the named units listed in ‘Summary of Key Quantities, Symbols and Units’ as appropriate. c. use SI base units to check the homogeneity of physical equations. d. show an understanding of and use the conventions for labelling graph axes and table columns as set out in the ASE publication Signs, Symbols and Systematics: The ASE Companion to 16 – 19 Science, 2000. e. use the following prefixes and their symbols to indicat e deci mal sub -multiples or multiples of both base and derived units: pico (p), nano (n), micro (μ), milli (m), centi (c), deci (d), kilo (k), mega (M), giga (G), tera (T). f. make reasonable estimates of physical quantities included within the syllabus. g. distinguish between scalar and vector quantities, and give examples of each. h. add and subtract coplanar vectors. i. represent a vector as two perpendicular components. j. show an understanding of the distinction between systematic errors (including zer o error) and random errors. k. show an understanding of the distinction between precision and accuracy. l. assess the uncertainty in a derived quantity by addition of actual, fractional, percentage uncertainties or by numerical substitution (a rigorous statistical treatment is not required). 1. SI Units (Le Système International d’Unités) A quantity is s omething that can be measured. Time is an example of a quantity. When physicists quote time intervals, they would use a numerical magnitude and a unit. An example of a measured time interval would be 9.58 seconds where “9. 58” represents a number (numerical magnitude) and “seconds” represents a unit in which time can be measured in. For a quantity to be meaningful, the unit is as important as the numerical magnitude: 9 seconds is certainly different from 9 hours. 1.1 Base Quantities and their Units The SI system of units consists of 7 base units in which all other units can be derived from using suitable combination of these base units. Base Quantities SI Base Units Name Symbol Length metre m Mass kilogram kg Time second s Amount of substance mole mol Temperature kelvin K Current ampere A *Luminous intensity candela cd Note: *Luminous intensity is not required in the syllabus.

1-2 1.2 Derived Quantities and their Units A derived quantity is linked to other quantities via a valid equation. Examples of derived quantities (with their units) Derived Quantities Equation Derived Units Area (A) A = l2 m2 Volume (V) V = l3 m3 Density ()  = m / V kg / m3 = kg m-3 Velocity (v) v = l / t m / s = m s-1 Acceleration (a) a = v / t (m s-1) / s = m s-2 Momentum (p) p = m × v (kg) (m s-1) The following table shows some derived units which are named after famous scientists. Derived Quantities Equation Derived Unit Derived Units in terms of base units Common Name Symbol Force p / t Newton N kg m s-2 Pressure F / A Pascal Pa kg m s-2 / m2 = kg m-1 s-2 Energy F × d Joule J kg m s-2 m = kg m2 s-2 Power E / t Watt W kg m2 s-2 / s = kg m2 s-3 Frequency 1 / t Hertz Hz 1 / s = s-1 Charge I × t Coulomb C A s Potential Difference E / Q Volt V kg m2 s-3 A-1 Resistance V / I Ohm  kg m2 s-3 A-2 (See APPENDIX for the various symbols for quantities.) A derived unit can be expressed in terms of products or quotients of base units.

1-3 Worked Example 1 Express the newton in terms of SI base units. Hint: Think of an equation with Force as the subject. Write down the equation. Solution: Since the Newton is the unit for force, think of an equation with force as the subject. The most obvious one is: F = m a Let [ ] denote ‘unit of’ Since, [F] = N, [m] = kg, [a]= m s-2 Thus, N = kg m s-2 Common Error: A common error in the solution is simply equating units to quantities. Eg. N = m a = kg x m s-2 This will be penalised because while the Newton is the unit for force; Newton is NOT force. Worked Example 2 (modified N02 P1 Q1) The drag force F experienced by a steel sphere of radius r dropping at speed v through a liquid is given by F = arv, where a is a constant. What is a suitable SI unit for a? Solution: Make a the subject first, a = F r v Let [ ] denote ‘unit of’  [a] = N m m s-1 = N m-2 s Another possible answer will be [a] = 12 2   sm smkg = kg m-1 s-1 {There are 2 possible answers here because question did not insist on SI base units.}

1-4 1.3 Concept of Homogeneity Consider the equation v = u + at from your O -level course. Since the equation is physically correct, it must be homogeneous or dimensionally consistent, then the three terms v, u and at must have the same SI base units: [v] = [u] = [at] [v] = m s-1 [u] = m s-1 [at] = m s-2 s = m s -1 From this, we can conclude that this equation is homogeneous. Note: Apart from checking if an equation is homogeneous, kno wing that the equation is homogeneous lets you work out the units of a n unknown term. For example, in the topic of Gravitational Fields (H2 Physics ), you will come across the equation F = GMm /r2, where G is a constant you have not come across yet, M and m are masses, F is a force and r is a distance. You can now work out the units of G, which is the gravitational constant (6.67 x 10-11 N m2 kg-2). 1.3.1 Limitations of Homogeneity It is a common error to conclude that a homogeneous equation is physically c orrect. For example, the following equations are homogeneous but physically wrong. Error Example 1. Incorrect unit-less coefficient v = 5u + at 2. Extra term v = u + at + as 3. Incorrect sign v = −u + at Worked Example 3 The motion of an object under constant acceleration a can be described by the equation v2 = u2 + 2as, where v is the final velocity and u is the initial velocity. Find the unit of s. Solution: If the equation is homogeneous, then [v2] = [u2] = [2as] = (m s-1)2 = m2 s-2. [s] = [as]/ [a] = (m2 s2 )/ (m s-2) = m Interesting History: Nov. 10, 1999: Metric Math Mistake Muffed Mars Meteorology Mission. A disaster investigation board reports that NASA’s Mars Climate Orbiter burned up in the Martian atmosphere because engineers failed to convert units from British Imperial to metric. Scan this QR code or go to http://www.wired.com/thisdayintech/2010/11/1110mars-climate-observer-report/

1-5 Worked Example 4 The flow of fluid can be described by the Bernoulli equation constant2 1 2  ghvP  , where P = pressure, ρ = density, v = velocity, g = gravitational acceleration, h = elevation. Determine if the equation is a correct one. Solution: If the equation is homogeneous,    ghvP      2 2 1 Term 1: p = force/ area [P] = kg m s-2 / m2 = kg m-1 s-2 Term 2:     2 2 1 v =  2v = kg m-3 (m s-1)2 = kg m-1 s-2 Term 3: [ρgh] = (kg m-3 )(m s-2)(m) = kg m-1 s-2 Since all 3 terms have the same SI base units, we can conclude that the equation is homogeneous (but we cannot comment on the correctness. Refer to 1.3.1) Remember: Presentation is very important. Many students will write h = m which is not correct. h is a quantity and m is a unit for the quantity. They are not the same. Prefixes for SI Units A unit can be made larger or smaller by placing a prefix before it. Multiplying Factor Prefix Symbol Example 10-12 pico p picometre (pm) – unit for wavelength of gamma-rays 10-9 nano n nanometre (nm) – unit for wavelength of light 10-6 micro  micrometre (m) – unit for size