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Units and Measurements 1 Units and Measurements MultipleChoiceQuestions (MCQs) 1 The number of significant figures in 0.06900 is (a)5 (b)4 (c)2 (d)3 K ThinkingProcess If the number is less than 1, the zero(s) on the right of decimal point and before the first non-zerodigitarenotsignificant. Ans. (b) In 0.06900, the underlined zeroes are not significant. Hence, number of significant figures are four (6900). 2 The sum of the numbers 436.32, 227.2 and 0.301 in appropriate significant figures is (a)663.821 (b)664 (c)663.8 (d)663.82 Ans. (b) The sum of the numbers can be calculated as 663.821 arithmetically. The number with least decimal places is 227.2 is correct to only one decimal place. Thefinalresultshould,thereforeberoundedofftoonedecimalplace i.e., 664. Note In calculating the sum, we should not confuse with the number of decimal places andsignificantfigures.Theresultshouldhaveleastnumberofdecimalplaces. 3 The mass and volume of a body are 4.237 g and 2.5 cm 3, respectively. The density of the material of the body in correct significant figures is − − − − (a)1.6048g cm 3 (b)1.69g cm 3 (c)1.7g cm 3 (d)1.695g cm 3 K ThinkingProcess In multiplication or division, the final result should retain as many significant figures as arethereintheoriginalnumberwiththeleastsignificantfigures. Ans. (c) In this question, density should be reported to two significant figures. 4.237g Density = = 1.6948 2.5 cm3 Asroundingoffthenumber,wegetdensity = 17. 1 2 (Class XI) Solutions 4 The numbers 2.745 and 2.735 on rounding off to 3 significant figures will give (a)2.75and2.74 (b)2.74and2.73 (c)2.75and2.73 (d)2.74and2.74 Ans. (d) Rounding off 2.745 to 3 significant figures it would be 2.74. Rounding off 2.735 to 3 significant figures it would be 2.74. 5 The length and breadth of a rectangular sheet are 16.2 cm and 10.1 cm, respectively. The area of the sheet in appropriate significant figures and error is (a) 164± 3 cm 2 (b) 163.62± 2.6 cm 2 (c) 163.6± 2.6 cm 2 (d) 163.62± 3 cm 2 K ThinkingProcess ∆ x If ∆ xiserrorinaphysicalquantity,thenrelativeerroriscalculated as . x Ans. (a) Given, length l =(16.2 ± 0.1)cm Breadth b =(10.1 ± 0.1)cm Area A=l × b =(16.2 cm) × (10.1 cm) = 163.62 cm2 Roundingofftothreesignificantdigits,area A = 164 cm2 ∆ ∆l ∆ A = + b =0.1 + 0.1 A l b 16.2 10.1 + = 1.01 1.62 = 2.63 16.2× 10.1 163.62 2.63 2.63 ⇒ ∆AA= × =163.62 × = 2.63 cm2 163.62 163.62 ∆A = 3 cm2 (By rounding off to one significant figure) ∴ Area, A =AA ± ∆ =(164 ± 3) cm2. 6 Which of the following pairs of physical quantities does not have same dimensional formula? (a) Work and torque (b) Angular momentumand Planck’s constant (c)Tensionandsurfacetension (d)Impulseandlinearmomentum − − Ans. (c) (a) Work =force × distance= [MLT2 ][L][ML2 T 2] − Torque=force × distance=[ML2 T 2] − (b) Angular momentum =mvr=[M][LT1][L] = [ML2 T 1] 2− 2 = E [ML T ] 2− 1 Planck’s constant = − =[ML T ] V [T1 ] − (c) Tension=force=[MLT 2] −2 force [MLT ] − Surface tension = = =[ML0 T 2 ] length [L] − − (d) Impulse =force × time=[MLT2 ][T]= [MLT 1 ] − − Momentum = mass × velocity =[M][LT1 ]= [MLT 1 ] 2 3 Note One should not be confused with the similar form tension in both the physical quantities-surface tension and tension. Dimensional formula for both of them is not same. 7 Measure of two quantities along with the precision of respective measuring instrument is − − A =2 5.ms1 ± 0.5 ms 1, B =0.10 s ± 0.01 s. The value of AB will be (a) (0.25± 0.08) m (b) (0.25± 0.5) m (c) (0.25± 0.05) m (d) (0.25± 0.135) m − − Ans. (a) Given, A =2.5 ms1 ± 0.5 ms 1, B =0.. 10s ± 0 01 s x= AB =(2 . 5 )( 010 . ) = 025 . m ∆ ∆ ∆ x = A + B x A B + =0. 5 +0. 01 = 0.. 05 0 025 = 0. 075 2. 5 010. 025. 025. ∆x =0.. 075 = 0 08 m,roundingofftotwosignificantfigures. AB =(..)0 25 ± 0 08 m 8 You measure two quantities as A = 1 0. m ± 0 2. m, B =2 .0 m ± 0.2 m. We should report correct value for AB as (a) 1.4 m± 0.4 m (b) 1.41 m± 0.15 m (c) 1.4 m± 0.3 m (d) 1.4 m± 0.2 m Ans. (d) Given, A =10..m ± 0 2 m, B =2.. 0m ± 0 2 m Let, Y= AB =(10 . )( 2 . 0 ) = 1414 . m Roundingofftotwosignificantdigit Y = 14. m ∆ ∆ ∆ Y = 1 A + B =1 0.2 + 0.2 = 0. 6 Y 2 A B 2 1.0 2.0 2× 2. 0 0. 6Y 0.. 6× 14 ⇒ ∆Y = = = 0. 212 2× 2. 0 2× 2. 0 Roundingofftoonesignificantdigit ∆Y = 0 2. m Thus,correctvaluefor AB =r +∆ r =1.4 ± 0.2m 9 Which of the following measurement is most precise? (a)5.00mm (b)5.00cm (c)5.00m (d)5.00km Ans. (a) All given measurements are correct upto two decimal places. As here 5.00 mm has the smallest unit and the error in 5.00 mm is least (commonly taken as 0.01 mm if not specified), hence, 5.00 mm is most precise. Note In solving these type of questions, we should be careful about units although their magnitudeissame. 10 The mean length of an object is 5 cm. Which of the following measurements is most accurate? (a)4.9cm (b)4.805cm (c)5.25cm (d)5.4cm Ans. (a) Given length l = 5 cm Now,checkingtheerrorswitheachoptionsonebyone,weget ∆l = − = 1 5 4 9. 0.1cm ∆l = − = 2 5 4 805. 0.195 cm 3 4 (Class XI) Solutions ∆l = − = 3 5.. 25 5 0 25 cm ∆l = − = 4 5.. 4 5 0 4 cm ∆ l Error 1 isleast. Hence,4.9cmismostprecise. 11 Young’s modulus of steel is 1.9× 10 11 N / m2. When expressed in CGS units of dyne/cm2, it will be equal to (1N = 10 5 dyne, 1 m2= 10 4 cm 2) (a) 1.9× 1010 (b) 1.9× 1011 (c) 1.9× 1012 (d) 1.9× 1013 Ans. (c) Given, Young’s modulus Y =19. × 1011 N/ m2 1N = 105 dyne Hence, Y =19. × 1011 × 10 5 dyne/m2 Weknowthat 1m = 100cm ∴ Y =19. × 1011 × 10 5 dyne/( )100 2 cm2 − =1.9 × 1016 4 dyne/cm2 Y =19. × 1012 dyne/cm2 Note While we are going through units conversion, we should keep in mind that proper relationbetweenunitsarementioned. 12 If momentum (p), area (A) and time ()T are taken to be fundamental quantities, then energy has the dimensional formula − − (a) [pA1 T 1 ] (b) [p2 AT] (c) [p A1/2 T] (d) [ pA1/2 T] Ans. (d) Given, fundamental quantities are momentum (p), area ()A and time ()T . Wecanwriteenergy E as E∝ pa A b T c E= kpa A A T c where k isdimensionlessconstantofproportionality. − − Dimensionsof EE=[ ] = [ML2 T 2 ] and []p = [MLT1 ] [A]= [L2 ] [T]= [T] [E]= [K]p[]a [A] b [T] c Puttingallthedimensions,weget − − MLT2 2= [MLT 1 ][L][T]a 2 b c + − + = MLTa2 b a a c Byprincipleofhomogeneityofdimensions, a = 1, 2b+ a = 2 ⇒ 2b + 1 = 2 ⇒ b = 1 2/ −a + c = − 2 ⇒ c= −2 + a = − 2 + 1 = − 1 − Hence, E= pA1/ 2 T 1 4 5 MultipleChoiceQuestions (MoreThanOneOptions) 13 On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is not correct? (a) y= asin2π t / T (b) y= asin vt a t 2πt 2 πt (c) y = sin (d) y= a 2sin − cos T a T T K ThinkingProcess We know that angle is dimensionless. Here, a is displacement and y is also displacement, hencetheyarehavingsamedimensions. Ans. (b, c) Now, by principle of homogeneity of dimensions LHS and RHS of (a) and (d) will be same andis L. For(c) [LHS] = L L − [RHS] = = LT 1 T [LHS] ≠ [RHS] Hence,(c)isnotcorrectoption. Inoption(b)dimensionofangleis []vt i.e., L ⇒ RHS = L.L = L2 andLHS =L ⇒ LHS ≠ RHS. So,option(b)isalsonotcorrect. 14 If P, Q, R are physical quantities, having different dimensions, which of the following combinations can never be a meaningful quantity? (a) (PQR− )/ (b) PQ− R (c) PQ/ R (d) (PR− Q2 )/ R (e) (RQP+ )/ K ThinkingProcess We should keep in mind that when two physical quantities are added or subtracted they shouldhavesamedimensions. Ans. (a, e) In this question, it is given that P, Q and R are having different dimensions, hence they cannot be added or subtracted, so we can say that (a) and (e) are not meaningful. We cannot say about the dimension of product of these quantities, hence (b), (c) and (d) may bemeaningful. Note In this question, we are certain about the quantity which is never meaningful but we shouldkeepinmindthatothersmayormaynotbemeaningful. 15 Photon is quantum of radiation with energy E= h ν, where ν is frequency and h is Planck’s constant. The dimensions of h are the same as that of (a)linearimpulse (b)angularimpulse (c)linearmomentum (d)angularmomentum Ans. (b, d) Weknowthatenergyofradiation, E= h ν 5 6 (Class XI) Solutions 2− 2 =[]E =[ML T ] = 2− 1 []h − [ML T ] []ν [T1 ] − Dimensionoflinearimpulse = Dimensionofmomentum = [MLT 1] Asweknowthatlinearimpulse JP= ∆ ⇒ Angularimpulse =τdt = ∆ L = Changeinangularmomentum Hence,dimensionofangularimpulse = Dimensionofangularmomentum − = [ML2 T 1]. ThisissimilartothedimensionofPlanck'sconstant h. 16 If Planck’s constant (h) and speed of light in vacuum ()c are taken as two fundamental quantities, which one of the following can, in addition, be taken to express length, mass and time in terms of the three chosen fundamental quantities? (a)Massofelectron ()me (b)Universalgravitationalconstant ()G (c)Chargeofelectron ()e (d)Massofproton ()mp Ans.
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