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Physics in operation…

Physics in operation…. Our universe works like a mechanical clock following a set of physical principles Given the laws of physics, the physical system will evolve and behave according to these laws. Chapter 1. Major Concepts. Physics and the Laws of Nature Units of Length, Mass, and Time

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Physics in operation…

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  1. Physics in operation…

  2. Our universe works like a mechanical clock following a set of physical principles Given the laws of physics, the physical system will evolve and behave according to these laws Chapter 1

  3. Major Concepts • Physics and the Laws of Nature • Units of Length, Mass, and Time • Dimensional Analysis • Converting Units • Significant figures and uncertainties • Order-of-Magnitude Calculations

  4. Almost everything you see are governed by physical laws

  5. Almost everything you see are governed by physical laws (cont.)

  6. Almost everything you see are governed by physical laws (cont.)

  7. Some attributes of physics • Knowing the physical laws allows us to know how the system behave/response/evolve • Physics is an empirical science – quantitative measurement of the physical systems is an integral part • Main objectives: to know how a given system evolve in time. This can be achieved if we know what are the fundamental laws that govern they system’s physical behavior • The observational results are used as the fundamental input in formulating a specific set of physical laws • Physics uses mathematics as it language • All physical theories has a range of validity • In physics we build models to represent a given phenomena so that it can be analysed easily • Example: A chicken is modeled to be a ball.

  8. Physics’ major branches • Classical mechanics (including relativity) • Statistical mechanics (including thermodynamics) • Electromagnetism (including optics) • Quantum mechanics

  9. Units and standards • Experiments must involve measurement of physical quantities • Fundamental quanties: • Mass: symbol: m, dimension: M, Standard Unit: kilogram • Length: symbol: l, dimension: L, Standard Unit: meter • Time: symbol: t, dimension: T, Standard Unit: second • Amount: symbol: n, dimension: 1, Standard Unit: mol • Temperature: symbol: T, dimension: T, Standard Unit: Kelvin. • All these quantities are to be standardised by comparing them against standard physical quantities that are invariable over time and are ‘constant’

  10. Example of S.I. • 1 kg is standardised against this • A second is standardised against this: 9,192,631,770 cycle of a cesium atomic clock

  11. Units of some familiar quantities • Speed, in unit ms-1 • Acceleration, in unit ms-2 • Force, in unit Newton, N. 1 N = 1 kgms-2 • Momentum, in unit Newton-second, Ns. 1 N = 1 kgms-1 • Energy, in unit Joule, 1 J = 1 Nm = 1 kgm2s-2

  12. prefix

  13. Scales in our Universe • Mass scales

  14. Time scales

  15. Length scales

  16. Flash movie: The scale of the universe

  17. Unit conversion • Must know how to convert from and fro a given unit to another • Example: • 1 kg = 103 g, 1 cm = 10-2 m • Aluminium’s density is • 2.70 g/cm3 = 2.70 (10-3 kg)/(10-2 m)3 • = 2.7 (10-3/ 10-6) kg/m3 • =2.7 x 103 kg/m3

  18. Example • Given 1 atomic mass u = 1.660 538 x 10-27kg • Mass of a carbon-12 nucleus: • 12 u = 12 x 1.660 538 x 10-27kg = 1.99846×10-26 kg • Try this: • 1 km = 103 m; 1 hour= 60 min = 60x60s = 3600 s; • Convert v = 50 km/h into m/s • Ans: 13.89 m/s

  19. Scientific notation • To easy numerical manipulation in calculation • Example: energy in a atom is of the order e ~ hc/l • h Planck constant ~ 10-34 Ns, c speed of light ~ 108m/s, l wavelength ~ 10-7 m • In S.I unit, e = (6.63 x 10-34 Js) x (3 x 108 m/s) / 10-7 m = 2 x 10-18 J

  20. Choice of unit • Choosing appropriate unit provide numerical ease of manipulation • Reflects also the relevant scale of the physics involved • Example: for system in atomic scale, convenient to work in units of eV and nm. • 1 Joule = (1/e) eV = 1/ (1.6 x 10-19 ) eV= 6.25 x 1018 eV •  hc = (6.63 x 10-34 Js) x (3 x 108 m/s) • = 6.63 x 10-34 x (6.25 x 1018 eVs) x • 3 x 108 x (109 nm/s) • = 1240 nmeV •  e = hc/l = 1240 nmeV / 100 nm = 12.4 eV (c.f e = 2 x 10-18 J in SI unit)

  21. Dimensional analysis • The dimension of a quantity Q is represented by [Q], i.e.,[m] = M. • All equations must be consistent in the dimension • Each single term in an equation must has the same dimension • [LHS] = [RHS] • e.g. v = u + a t is dimensionally consistent • [v] = LT-1; [u] = LT-1; [a] = LT-2 ;[t] = T • e.g. v2 = u + a t is dimensionally inconsistent

  22. Dimensions of Some Common Physical Quantities

  23. Dimensional analysis with power law • Given the angular acceleration a of an object executing uniform circular motion is dependent on the linear velocity v and its position from the circle’s center, r. Determine the easiest possible relation between {v, r} and a.

  24. Solution • The most general power law relation is • a  vn rm a = k vn rm(k dimensionless proportinal constant) • [a] = L/T2; [v]n = (L/T)n; [r]m = (L)m • Equating both sides • L/T2 = (L/T)n(L)m = (Ln+m)(T)-n •  n + m =1; n = 2 • m = 1 – n = -1 •  a = k v2 r -1 = k v2/r

  25. Quick Quiz • True or false: Dimensional analysis can provide the numerical value of the proportional constant in the formula. • Ans: Of course it is false.

  26. Every single measurement always contains uncertainty • The value of a quantity measured with an apparatus is only accurate up to certain limit, depending on factors such as the quality of the apparatus, the way the experiment is carried out, sizes of the measurement sample taken etc. • In other words, every measured value must contain uncertainty due to the limited accuracy of the apparatus. • How is the uncertainty of the measured value be quantified?  Use significant figure (SF) and uncertainty ( D) • In many occasion, we would use the term “accuracy” for “uncertainty” interchangeably to refer to the same thing.

  27. Example of significant figure and decimal places • The number of SF in “1500 g” is 2, not 4; • The number of SF in “1500.0 g” is 5 • The number of SF in “0.00300 kg” is …3 • To tell how many SF is, express the value in scientific notation • 0.00300 kg = 3 x 10-3 kg  1 SF, 3 decimal places (DP) • 0.000 23 = 2.3 x 10-4 2 SF, 5 DP • 2.300001 = 2.300001 x 100  7 SF, 6 DP • 1500 g = 1.5 x 10-3 g  2 SF, 0 DP.

  28. For example, consider measuring the sides of a piece of diskette with a ruler that can measure only up to the accuracy of  0.1 cm. The resultant measurement is expressed as 5.5  0.1 cm and 6.4  0.1 cm. In this example, the results of the measurement contain only 2 significant figures, 1 DP. In general, the larger the number of significant figure of a measured value is, the higher is the accuracy. Example of quantifying the uncertainty in a measured value

  29. Measurement using a more accurate apparatus • As a comparison, a measurement made using a Vernier caliper has an accuracy of 0.01mm = 0.001cm. • The sides of the diskette measured with it will take the form 5.501  0.001 cm and 6.403  0.001 cm, which have 4 significant figure. • The measurement made with the Vernier caliper is higher than that made with a ruler. This can simply be inferred by counting the number of S.F. of the same quantity (the sides of the diskette) using both apparatus.

  30. How to calculate error in a derived quantity • Say a q quantity is derived from some other quantities a, b, c,…, which carry with them uncertainties Da, Db, Dc, …, i.e, q = q (a, b, c,…) • The uncertainty in a, b, c, … will propagate to the derived quantity q. We hence expect that the uncertainty in q must be a function of Da, Db, Dc, … • We wish to derive Dq = Dq(Da, Db, Dc, …). • In general, the uncertainty in q, Dq, can be determined from the function of q on a, b, c…

  31. What is the uncertainty in the area of the diskette, A? a = (5.5  0.1) cm; b = (6.4  0.1) cm, a = b = 0.1 cm A = ab = (5.5 x 6.4) cm2 = 34.02 cm2 The error / uncertainty in A is given by (A/A)2 = (a/a)2 + (b/b)2 (A/A) = [(a/a)2 + (b/b)2]½ ≈ a/a + b/b (assuming a/a,b/b << 1)  A ≈ (a/a + b/b)A = (0.1/5.5 + 0.1/6.4)x5.5x6.4 cm2 = 1.17 cm2 The resultant uncertainty in the derived quantity, A, must be rounded up to only 1 SF: A = 1 cm2 The expression of the value of A, namely, the number of decimal point (DP) must be consistent with that of A. Hence, the area A must be expressed as (34  1) cm2 (2 S.F, 0 DP) but not A = (34.02  1.17) cm2 or A = (34.0  1.2) cm2 Example of determining the accuracy of a derived quantity

  32. Comparison with the values measured with Vernier caliber • As a comparison, let’s estimate the uncertainty in A = ba,where the sides a and b are measured with Vernier caliber. • In this case, a = (5.501  0.001) cm; b = (6.403  0.001) cm, a = b = 0.001 cm. • Let A = ba = (5.501 x 6.403) cm2 = 35.2229 cm2 • A ≈ (a/a + b/b)A =(0.001/5.501+0.001/6.403)x35.2229 cm2 = 0.011904 cm2 • As in the previous example, the uncertainty is usually rounded up to only 1 SF, i.e., A = 0.01 cm2 • Hence the area measured by the Vernier caliber, based on the measured sides a and b are A = (35.22  0.01) cm2 (4 SF, 2 DP) c.f A = (34  1) cm2 (2 S.F, 0 DP) with ruler

  33. Uncertainty in addition and subtraction • How to calculate the uncertainty in X = a - b or a + b, where a and b carry a uncertainty a, b? • Take the example measured by the ruler, a = (5.5  0.1) cm; b = (6.4  0.1) cm, a = b = 0.1 cm • Let X = b  a = 11.9or 0.9 cm • The error (or uncertainty) in X is given by • X = [(a)2 + (b)2]½ ≈ a + b = (0.1 + 0.1) = 0.2 cm (1 SF) • Hence X is expressed as X = 11.9  0.2 cm or 0.9  0.2 cm.

  34. Estimation and order of magnitude calculation • In many occasion we don’t really need to know the exact value of a quantity. A rough order of magnitude estimation could be enough to provide us some good insight of the size of an effect we wish to access. • We usually can carry out some educated guesses on the order of magnitude of a desired quantity before proceed with a detailed full calculation. • This order of magnitude estimation is often very useful in physics problem. It give us a good sense of the scale of the problem under consideration. • Usually in the order of magnitude estimation, we round up a quantity to the closest order, i.e., to the accuracy within a factor of 10. • The symbol “~” is used in the estimation calculation. • Example: 0.0086 ~ 10-2, 0.0021 ~ 10-3, 720 ~ 102

  35. Estimation and order of magnitude calculation (cont.) • For example, the effect of the centrifugal acceleration acting on us due to the rotational motion of Earth about its own axis can be estimated and compared with the gravitational acceleration quite easily: • ac = w2R = (2p/24x3600s)2 (6400 km) = [2x3.142 / (2.4x101 3.6x103)]2 (6.4x106 m) ~ 100. 10-8  106 = 10-2 m/s2, to be compared with gravitational acceleration g ~ 101 m/s2. • We see that ac << g. The effect from centrifugal acceleration is tiny compared with that from gravitational pull.

  36. Example • Estimate how many times a person breath in his/her entire life span. • Estimate the life span ~ 70 year. • Roughly, a person breath 10 times in a minute • Hence the total number of breathing in one’s life is ~ 70 yr x (10 times/min) = 70 x (365 x 24 x 60 min) x (10 times/min) = 7x101 x (4x102 x 2x101 x 6x102 min) x(101times/min) ~ 101+2+1+2+1 ~ 107 times • (108 is also an acceptable answer) • What is important here is not the exact value but only the order of magnitude.

  37. Example Can you estimate how long it takes to walk from Penang to KL?

  38. Walking to KL from Penang • Estimate distance ~ 400 km • Distance of each step ~ 0.5 m • Hence number of steps in 400 km is • ~ 400 km / (0.5 m/step) • = 4x105 m / (0.5 m / step) = 2 x105 steps • Estimation of duration: • Assume your walking speed is ~ 5 km/ h • Hence the total duration (without resting) is ~ 400 km / (5 km/h) = 80 h • If resting time is taken into account, where you walk 12 hours per day and rest 12 hours per day, • total duration would be 2 x 80 h = 160 j ~ 6.7 days ~ 7 days.

  39. Try to estimate it yourself • What is the total number of teeth in the whole USM campus at any time? • Hint: How many teeth you have?

  40. Summary • The general definition of physics and some of the conceptual tools needed to begin its study • Know the three most common basic physical quantities in physics and their units • Know how to determine the dimension of a quantity and perform a dimensional check on any equation • Be familiar with the most common metric prefix • be able to convert quantities from one set of units to another • Express calculated values with the correct number of significant figures and their uncertainty • Be able to perform quick order of magnitude calculations

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