MEASUREMENTS AND UNCERTAINTIES

Transcription

1 Physics for the IB Diplom Exm Preprtion Guide MEASUREMENTS AND UNCERTAINTIES This chpter covers the following topics: Fundmentl nd derived units Significnt figures nd scientific nottion Order-of-mgnitude estimtes Uncertinties, grdients nd intercepts Lineristion of grphs Vectors nd sclrs Rndom nd systemtic errors. Units It is fscinting fct tht ll physicl quntities hve units tht cn be expressed in terms of those for just seven fundmentl quntities. DEFINITIONS FUNDAMENTAL UNITS The seven fundmentl quntities in the S.I. system (the IB syllbus uses only the first six) nd their units re: Time Length Mss Temperture Quntity of mtter Electric current Luminous intensity second (s) meter (m) ilogrm (g) elvin (K) mole (mol) mpere (A) cndel (cd) DERIVED UNITS All other quntities hve derived units, tht is, combintions of the fundmentl units. For exmple, the derived unit for force (the newton, N) is obtined using F = m to be g m s nd tht for electric potentil difference (the volt, V) is obtined using W = qv to be J Nm gms m 3 = = = gm s A. C As As SIGNIFICANT FIGURES There is difference between stting tht the mesured mss of body is g nd sying it is 83.6 g. The impliction is tht the uncertinty in the first mesurement is ±. g nd tht in the second is ±. g. Tht is, the first mesurement is more precise it hs more significnt figures (s.f.). When we do opertions with numbers (multipliction, division, powers nd roots) we must express the result to the sme number of s.f. s in the lest precisely nown number in the opertion (Tble.). Tble. Number Number of s.f. Scientific nottion or just Zeros in front do not count but zeros t the end in deciml do count Zeros t the end in n integer do not count. 3.4

2 Physics for the IB Diplom Exm Preprtion Guide Mesurements nd Uncertinties Thus the inetic energy of mss of.4 g ( s.f.) moving t 4.6 m s (3 s.f.) is given s E K = = J 6 =.6 J ( s.f.). Similrly, the ccelertion of body of mss g ( s.f.) cted upon by net force of 55 N (3 s.f.) is given s 55 = m s ( s.f.). TEST YOURSELF. The force of resistnce from fl uid on sphere of rdius r is given by F of the sphere nd η is constnt. Wht re the units of η? TEST YOURSELF. = 6πηrv, where v is the speed The rdius R of the fi rebll t seconds fter the explosion of nucler wepon depends only on the energy E relesed in the explosion, the density ρ of ir nd the time t. Show tht the quntity Et units of m 5 nd hence tht R Et 5 fter.5 s. (Te ρ =.gm 3.) ρ. Clculte the energy relesed if the rdius of the firebll is 4 m ρ hs. Uncertinties DEFINITIONS RANDOM UNCERTAINTIES Uncertinties due to the inexperience of the experimenter nd the diffi culty of reding instruments. Ting n verge of mny mesurements leds to more ccurte result. The verge of n mesurements x, x,, x n is _ x x = + x + + x n n. SYSTEMATIC UNCERTAINTIES Uncertinties due minly to incorrectly clibrted instruments. They cnnot be reduced by repeted mesurements. ACCURATE MEASUREMENTS Mesurements tht hve smll systemtic error. PRECISE MEASUREMENTS Mesurements tht hve smll rndom error. Nture of Science. A ey prt of the scientii c method is recognising the errors tht re present in the experimentl technique being used, nd woring to reduce these s much s possible. In this section you hve lerned how to clculte errors in quntities tht re combined in dif erent wys nd how to estimte errors from grphs. You hve lso lerned how to recognise systemtic nd rndom errors. No mtter how much cre is ten, scientists now tht their results re uncertin. However, they need to distinguish between inccurcy nd uncertinty, nd to now how coni dent they cn be bout the vlidity of their results. The serch to gin more ccurte results pushes scientists to try new ides nd rei ne their techniques. There is lwys the possibility tht new result my coni rm hypothesis for the present, or it my overturn current theory nd open new re of reserch. Being wre of doubt nd uncertinty re ey to driving science forwrd. Normlly we express uncertinties to just one signii cnt i gure. However, if more sophisticted sttisticl nlysis of the dt hs ten plce there is some justii ction for eeping two signii cnt i gures. In generl, for quntity Q we hve Q = Q ± ΔQ mesured vlue bsolute uncertinty, ΔQ Q = frctionl uncertinty, ΔQ Q % = percentge uncertinty

3 Physics for the IB Diplom Exm Preprtion Guide. Uncertinties As n exmple, consider mesurement of the length of the side of cube, given s 5 ± mm. The 5 mm represents the mesured vlue of the length nd the ± mm represents the bsolute uncertinty in the mesured vlue. The rtio 5 =.4 is the frctionl uncertinty in the length, nd 5 % = 4% is the percentge uncertinty in the length. Suppose quntities, b nd c hve been mesured with bsolute uncertinties, respectively, of Δ, Δb nd Δc. If we use, b nd c to clculte nother quntity Q, these uncertinties will result in n uncertinty in Q. The (pproximte) rules for clculting the uncertinty ΔQ in Q re: If Q = ± b ± c, then ΔQ = Δ + Δb + Δc. Tht is, for ddition nd subtrction, dd the bsolute uncertinties. If Q = b c, then Q = + b + c. Tht is, for multipliction nd/or division dd the Q b c frctionl uncertinties. If Q = n m b, then Q = n + m b. In prticulr, if Q = b or = Q b Q b, then Q b = + Q b. Model Answer. The volume of cylinder of bse rdius R nd height H is given by V = R H. The volume of cylinder is mesured to % nd height to 4%. Estimte the percentge uncertinty in the rdius. First solve for the vrible whose uncertinty we wnt to estimte: R R V H Hence = + ( 4) 7% R V H = + =. V =. πh TEST YOURSELF.3 V The resistnce of lmp is given by R =. The uncertinty in the voltge is 4% nd the uncertinty in I the current is 6%. Wht is the bsolute uncertinty in clculted resistnce vlue of 4 Ω? TEST YOURSELF.4 Ech side of cube is mesured with frctionl uncertinty of.. Estimte the percentge uncertinty in the volume of the cube. TEST YOURSELF.5 The period of oscilltion of mss m t the end of spring of spring constnt is m given by T = π. Wht is the percentge uncertinty in the period if m is mesured with percentge uncertinty of 4% nd the with percentge uncertinty of 6%? Avoid the common miste of sying tht the uncertinty is (4% + 6%) 3%. 3

4 Physics for the IB Diplom Exm Preprtion Guide Mesurements nd Uncertinties Error brs Suppose tht we wnt to plot the point (3. ±., 5. ±.) on set of x nd y xes. First we plot the point with coordintes (3., 5.) nd then show the uncertinties s error brs (Figure.). The horizontl error br will hve length. =. nd the verticl will hve length. =.4. y x DEFINITIONS BEST-FIT LINE The curve or stright line tht goes through ll the error brs; n exmple is shown in Figure.. Note tht line my be stright or curved. Figure. Finding slopes To ind the slope (or grdient) of curve t prticulr point (here t x =. m), drw the tngent to the curve t tht point (Figure.3). Choose two points on the tngent tht re s fr prt s possible. Note in this cse tht the units on the horizontl xis re m, nd tht the slope hs negtive vlue. 6.. Slope =.. =.4 Vm volt m V / volt Figure Figure. x / m.5 Estimting res under curves To estimte the re under the blc curve in Figure.4, drw stright line (red) from the point (, 6) to the point (4,.5). It is esy to clculte the re of the trpezium under the stright line, s (6 +.5) 4 = 5.. Now estimte the number of smll squres in the spce between the stright line nd the curve, nd subtrct this from the totl, to give the re under the curve. There re bout 53 squres. Ech one hs re of.5.5 =.65 squre units, so the re between the curve nd the stright line is bout = 3.3. So the re under the curve is bout =.7 squre units. y Figure.4 x 4

5 Physics for the IB Diplom Exm Preprtion Guide. Uncertinties Getting stright-line grphs If we now the reltionship between two vribles we cn usully rrnge to plot the dt in such wy tht we get stright line. Ber in mind tht the stndrd eqution of stright line is y = m x + c grdient verticl intercept where we plot the vrible y on the verticl xis nd the vrible x on the horizontl xis. If the stright line goes through the origin (c = ), we sy tht y is proportionl to x. If the best-it line is not stright or if it does not go through the origin, then either of these resons is suicient to clim tht y is not proportionl to x. m As n exmple, consider the reltionship T = π for the period T of mss m undergoing oscilltions t the end of spring of spring constnt. Compre this eqution nd the generl stright-line eqution: T y π = π = constnts m x By identifying T y nd m x we get the eqution of stright line, y = π x. So if we plot T on the π verticl xis nd m on the horizontl xis we should get stright line whose grdient is. Alterntively, we my write: 4π T = m y 4π = constnts x π By identifying T y nd m x we get the eqution of stright line, = 4 y x. So if we plot T on 4π the verticl xis nd m on the horizontl xis we should get stright line whose grdient is. A diferent procedure must be followed if the vribles re relted through power reltion such s F = r n, where the constnts nd n re unnown. Ting nturl logs (or logs to ny other bse), we hve: ln F = ln + n ln r y = ln + n x nd so plotting ln F versus ln r should give stright line with grdient n nd verticl intercept ln. λt A vrition of this is used for n exponentil eqution such s A = A e, where A nd λ re constnts. Here we cn te the logs of both sides to get ln A = ln A λt, nd so: ln A = ln A λ t y = ln A λ x 5

6 Physics for the IB Diplom Exm Preprtion Guide Mesurements nd Uncertinties Plotting ln A on the verticl xis nd t on the horizontl then gives stright line with grdient λ nd verticl intercept ln A. TEST YOURSELF.6 Tble. Copy Tble. nd fill in the bln entries. Eqution Constnts Vribles to be plotted to give stright line P = T v = u + t v = s F V T qq = r = ω x u,, q, q ω q =, q r I = I λ = 4π GM R 3 = G, M e T h mqv I, h, m, q F = v + bv, b E = mω A x m, ω, A + = f u v f Grdient Verticl intercept TEST YOURSELF.7 Stte wht vribles must be plotted so tht we get stright line for the reltion d = ch.8, where c is constnt. Estimting uncertinties in mesured quntities Useful simple rules re for estimting the uncertinty in mesured quntity is: For nlogue meters, use hlf of the smllest scle division. For exmple, for n ordinry meter rule the smllest scle division is mm nd so the uncertinty is ±.5 mm. If this is used, for exmple, to mesure the length of rod, this uncertinty pplies to the position of ech end of the rod, for totl uncertinty of ± mm in the rod s length. For digitl meters, use the smllest division. For exmple, with digitl voltmeter tht cn red to the nerest hundredth of volt, te the uncertinty s ±. V. For n mmeter tht cn red to the nerest tenth of n mpere, te the uncertinty s ±. A. 6

7 Physics for the IB Diplom Exm Preprtion Guide. Uncertinties TEST YOURSELF.8 Estimte the reding nd the uncertinty in ech of the instruments in Figure.5. b cm Figure.5 Annotted Exemplr Answer. The period of pendulum is mesured to be T = (. ±.5) s. Clculte the vlue of T, including its uncertinty. [3] T =. = 4.84 s ΔT =.5 =.5 s So T = (4.84 ±.5) s. The finl nswer gins no mrs becuse the wrong method ws used to find ΔT. One wy to spot errors lie this is to s if the finl nswer is sensible. Here the uncertinty is given to s.f. in the third nd fourth deciml plces, when the vlue hs two deciml plces, so the nswer must be wrong. The correct nswer is T = (4.8 ±.) s. The vlue of T is correct, nd with the correct units. The vlue of ΔT is.5 s, but you cnnot squre the uncertinty in T to find ΔT, the uncertinty in T. Me sure you now how to find frctionl uncertinties when there re powers, nd remember the protocol for the number of significnt figures in the uncertinty. Here the frctionl uncertinty in T is ΔT T =.5. =.. (rounded to s.f. s the uncertinty is greter thn %). The power in T is, so multiply the frctionl uncertinty in T by to find the frctionl uncertinty in T, tht is,. =.4. So ΔT = s. /3 Uncertinty in the mesured vlue of grdient (slope) To i nd the uncertinty in the grdient of the (stright) best-i t line, drw lines of mximum nd minimum grdient. You must judge these by eye, ting into ccount ll error brs, not just those of the i rst nd lst dt points. Clculte these two grdients, m mx nd m min. A simple estimte of the uncertinty in the grdient is then mmx m min. TEST YOURSELF.9 Electrons tht hve been ccelerted through potentil difference V enter region of mgnetic field B, m where they re bent into circulr pth of rdius r. Theory suggests tht r = qb V, where q is the electron s chrge nd m is its mss. Tble.3 shows vlues of the potentil difference V nd rdius r obtined in n experiment. 7

8 Physics for the IB Diplom Exm Preprtion Guide Mesurements nd Uncertinties Tble.3 Rdius r / cm ±. cm Potentil difference V / V r / cm ± ± ± ± 6. 9 ± Explin why grph of r ginst V will result in stright line. b Stte the slope of the stright line in in terms of the symbols m, q, B. c Copy Tble.3 nd in the right-hnd column insert vlues of the rdius squred, including its uncertinty. Figure.6 shows the dt points plotted on set of xes. d Drw error brs for the ll the dt points. e Drw best-fit line for these dt points. f Clculte the grdient of the best-fit line, including its uncertinty r / cm V /V 8 Figure.6 The mgnetic field used in this experiment ws B =.8 3 T. q g Clculte the vlue tht this experiment gives for the chrge-to-mss rtio m of the electron. Include the uncertinty in the clculted vlue..3 Vectors nd sclr quntities DEFINITIONS VECTOR A physicl quntity tht hs both mgnitude nd direction. It is represented by rrows. The length of the rrow gives the mgnitude of the vector. The direction of the rrow is the direction of the vector. Exmples of vectors re displcement, velocity, ccelertion, force, momentum nd electric/grvittionl/mgnetic fields. SCALARS A physicl quntity with mgnitude but not direction. A sclr cn be positive or negtive. Exmples re distnce, speed, mss, time, wor/energy, electric/grvittionl potentil nd temperture. 8

9 Physics for the IB Diplom Exm Preprtion Guide.3 Vectors nd sclr quntities Adding vectors: hve nd b strt t the sme point, O (Figure.7). Drw the prllelogrm whose two sides re nd b. Drw the digonl strting t O. b b b Subtrcting vectors: hve nd b strt t the sme point, O (Figure.7b). To ind b drw the vector from the tip of to the tip of b. O b + O b Components of vectors b c R Figure.7 Addition; b subtrction. F sin F T cos T mg sin mg cos F cos T sin mg Figure.8 Components of vectors. The component djcent to the ngle θ involves cos θ nd tht opposite to θ involves sin θ. Drw the forces. Put xes. Get components. Choose s one of your xis the direction in which the body moves or would move if it could. TEST YOURSELF. A river is 6 m wide. A bot cn trvel t 4. m s with respect to the wter nd the current hs speed of 3. m s with respect to the shore, directed to the right (Figure.9). The bot is rowed in such wy s to rrive t the opposite shore directly cross from where it strted. Clculte the time ten for the trip. required pth of bot 6 m current Figure.9 9

10 Physics for the IB Diplom Exm Preprtion Guide Mesurements nd Uncertinties.4 Order-of-mgnitude estimtes Tbles.4,.5 nd.6 give typicl vlues for vrious distnces, msses nd times. You re not expected to now these by hert but you must hve generl ide of such sizes, msses nd durtions. Tble.4 Tble.5 Length / m Mss / g Rdius of observble universe 6 The universe 53 Distnce to the Andromed glxy The Mily Wy glxy 4 Dimeter of the Mily Wy glxy The Sun 3 Distnce to Proxim Centuri (str) 6 The Erth 4 Dimeter of solr system 3 Boeing 747 (empty) 5 Distnce to the Sun An pple. Rdius of the Erth 7 A rindrop 6 Size of cell 5 A bcterium 5 Size of hydrogen tom Mss of smllest virus Size of n verge nucleus 5 A hydrogen tom 7 Plnc length 35 An electron 3 Tble.6 Time / s Age of the universe 7 Time of trvel for light from nerby str (Proxim Centuri) 8 One yer 7 One dy 5 Period of hertbet Period of red light Time of pssge of light cross n verge nucleus Plnc time TEST YOURSELF. Estimte the weight of n pple. TEST YOURSELF. Estimte the number of seconds in yer. TEST YOURSELF.3 Estimte the time ten by light to trvel cross nucleus.