DHS 01 Measurements (Lecture Notes & Tutorial)

Dunman High School (Senior High Physics) 1 Topic 1 - Measurement Guiding Questions  How are the standards for measurements established?  Why is uncertainty inherent in all measurements?  How can uncertainties be estimated? How can they be reduced if necessary?  Why does the uncertainty of a measurement matter?  How is the skill of making estimate s of physical quantities useful and how can this skill be developed? Contents  Physical quantities & SI Units  Scalars and vectors  Errors and uncertainties Learning Outcomes Students should be able to: (a) recall the following base quantities and their units : mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol). (b) express derived units as products or quotient s 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 indicate decimal 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 physi cal 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 s ystematic errors (including zero errors) and random errors. (k) show an understanding of the dist inction between precision and accuracy. (l) assess the uncertainty in a derived quantity by addi tion of actual, fractional or percentage uncertainties or numerical substitution (a rigorous statistical treatment is not required). 1 Physical Quantities and SI Units Any scientifically measurable quantities are called physical quantities. e.g. temperature, force, current, pressure, mass, energy, etc. All physical quantities consist of a numerical value and a unit. e.g. mass, m = 65.0 kg The Summary of some Quantities and Units used in the A-Level examination is given in Appendix 1. Physical Quantity Numerical value unit

Dunman High School Year 5 Physics 2024/2025 2 In very much the same way that languages have deve loped in various parts of the world, different units of measurements have evolved. Just as languages can be translated from one to another, units of measurement can be converted between systems. For standardisation, it is much better to have just one system of units. Scientists use the Système International (SI) which is based on the metric system. LO(a) recall the following base quantities and their units: mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol). Physical quantities are divided into base quantities and derived quantities. Base quantities are the seven physical quantities of the SI system by which all other physical quantities are defined. They are: mass, length, time, temperature, amo unt of substance, electric current and luminous intensity. The SI base units are a choice of seven well-de fined units which by convention are regarded as dimensionally independent: kilogram (kg), metre (m), se cond (s), ampere (A), kelvin (K), mole (mol). Candela (cd) is not in the syllabus. LO (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’1 as appropriate.   Base units are the seven units of the SI system, related to the base quantities, whose magnitude is defined without referring to other units. Derived units are units that are derived from base units and can be expressed in terms of products and quotients of base units. Derived quantities are physical quantities that are derived from base quantities and can be expressed in terms of products and quotients of base quantities. E.g. The physical quantity speed, v, may be expressed in terms of the quantities distance x and time t by the defining equation v = x/t . x and t are base quantities, with SI base units metre m and second s respectively. Speed v is then a derived quantity, with the derived unit of speed metre-per-second, m s–1. Physical Quantities Base quantities Derived quantities

Dunman High School Year 5 Physics 2024/2025 3 Some derived units are given names for convenience. Quantity Derived Units Special name Symbol Volume m  m  m = m3   Velocity m  s = m s–1   Force kg  (m  s  s) = kg m s–2 newton N Work done kg m s–2  m = kg m2 s–2 joule J Power (kg m2 s-2)  s = kg m2 s–3 watt W Some quantities are dimensionless / unitless, such as: 1. all numbers, e.g. 2, 1 2 ,  , e 2. trigonometrical functions, e.g. sine, cosine, tangent 3. all logarithmic functions, e.g. log x, ln 4. powers, e.g. y x 10 , the ratio y x must be unit-less. If x has a unit, then the unit of y must have the same unit as that of x. 5. Unit-less physical constants: e.g. refractive index of glass, relative density of a liquid. Practice 1: Derived quantity Obtained from Derived unit Special name Density mass / volume k g m-3 - Frequency Number of cycles per unit time s −1 Hz Pressure force / area kg m −1 s−2 Pa Charge current x time A s C Potential difference work done / charge Kg m 2 A−1 s−3 V LO (c) Use SI base units to check the homogeneity of physical equation. An equation is homogeneous if the base units of all the terms in the equation are the same. Every term on both sides of the equal sign of an equat ion should have the same units, for the equation to be called dimensionally consistent or homogeneous. This is just plain common sense, as when Z = X + Y , we expect all quantities, Z, X and Y to represent the same item. Consider the equation: s = ut + ½at2 Unit of s = m Unit of ut = (unit of velocity)  (unit of time) = m s1  s = m Unit of ½at2 = m s2  s2 = m Since the units of every term on both sides of the equation are the same, the equation is homogeneous. Next consider the equation: v = u 2 + 2as2

Dunman High School Year 5 Physics 2024/2025 4 Unit of v = m s1 Unit of u2 = (unit of velocity)2 = (m sl) 2 = m2 s2 Unit of as2 = m s2  m2 = m3 s2 The equation is not homogeneous. Checking the homogeneity of an equation using base units is a powerful way of establishing if the physical equation is reasonable. It narrows the numerous combinations that may exist. A physically correct equation must always be homogeneous. However, a homogeneous equation need not be physically correct. Why? There are two basic reasons: (1) The value of the dimensionless factor may be incorrect. e.g. 23mvE  w h e r e E = kinetic energy The coefficient 3 is incorrect! The value should be 2 1 instead. (2) Missing or extra terms that may have the same unit. e.g. mghmvE  2 2 1 where E = kinetic energy There is an extra term mgh, which happens to have the same unit as kinetic energy. This is an extra term. Practice 2: The period T of a simple pendulum is thought to depend on its length l, its mass m and the acceleration due to gravity g according to the equation T = k l x my gz , where k is a dimensionless constant (i.e. a constant with no unit). Determine the indices x, y and z. Base units of T = s Base units of k lx my gz = mx kgy (m s2)z = mx+z kgy s2z Comparing the indices of kg : y = 0 s: 2z = 1  z = ½ m: x + z = 0  x =z = ½ A possible equation is: T