Mxmum Mrks: 18 Totl tme llowed: 35 mn Quz: Expermentl Physcs Lb-I Nme: Roll no: Attempt ll questons. 1. In n experment, bll of mss 100 g s dropped from heght of 65 cm nto the snd contner, the mpct s clled crter. Fve students mesured the dmeter of the crter from the sme heght. The dt for ech student s shown n Fgure (1), whch student mde the most precse mesurement? [3] No. of trls No. of trls 15 10 5 15 10 5 3.7 4.7 5.7 6.7 Crter dmeter (cm) () 3.7 4.7 5.7 6.7 Crter dmeter (cm) (c) No. of trls No. of trls 15 10 5 15 10 5 3.7 4.7 5.7 6.7 Crter dmeter (cm) (b) 3.7 4.7 5.7 6.7 Crter dmeter (cm) (d) No. of trls 15 10 5 3.7 4.7 5.7 6.7 Crter dmeter (cm) (e) FIG. 1: Expermentl dt for crter formton. Soluton: The correct nswer s (). Snce type-a uncertnty s evluted sttstclly nd cn be mnmzed by repetng the experment mny tmes, therefore the spred of the Gussn dstrbuton ssocted wth type-a uncertntes should be s thn s possble. Thus, the wdth of the crter dmeter vs number of trls grph must hve lest spred. The only choce tht shows ths chrcterstc s choce (). Dte: Tuesdy, December 2, 2014. 1
Mxmum Mrks: 18 Totl tme llowed: 35 mn 2. The fgure bove represents log-log (to the bse 10) plot of vrble y versus vrble x. The orgn represents the pont x 1 nd y 1. Whch of the followng gves the pproxmte functonl reltonshp between y nd x? [3] FIG. 2: Log-log plot of vrble y versus vrble x. () y c x. (b) y 1x + c. 2 (c) y 6x + c. (d) y cx 2. (e) log y c + 5 log x. Soluton: The correct nswer s (). A functon of the form y cx m wll pper s strght lne on log-log plot. Here m s the slope of the lne nd c s the y vlue correspondng to x 1. Therefore, log 10 (y) log 10 (cx m ), log 10 (c) + m log 10 x. Dte: Tuesdy, December 2, 2014. 2
Mxmum Mrks: 18 Totl tme llowed: 35 mn The vlue of the slope cn be found out, m log 10(100) log 10 (10) log 10 (300) log 10 (3), 2 1 2.477 0.477 1 2. The reltonshp between y nd x wll tke the form, log 10 (y) log 10 (c) + 1 2 log 10 T, or y c x 1/2. 3. The volume V of rectngulr block s determned by mesurng the length l x, l y nd l z of ts sdes. From the sctter of the mesurements stndrd uncertnty of 0.01% s ssgned to ech dmenson. Wht s the frctonl uncertnty n V, f, (1) The sctter s due to uncertntes n settng nd redng the mesurng nstrument. (2) If t s due to temperture fluctutons? [3] () 0.2% nd 0.3% respectvely. (b) 0.02% nd 5% respectvely. (c) 0.02% nd 0.03% respectvely. (d) 0.03% nd 0.02% respectvely. (e) None of the bove Soluton: The correct nswer s (c). (1) The stndrd uncertnty n ech dmenson s 0.01%. The volume of rectngulr block s, V l x l y l z. The uncertnty ffects the three sdes ndependently. Hence, the stndrd uncertnty n V cn be clculted through the Tylor seres pproxmton, ( V ) 2 ( ) 2 ( ) 2 V V V l x + l y + l z, l x l y l z (ly l z l x ) 2 + ( lx l z l y ) 2 + ( lx l y l z ) 2. Dte: Tuesdy, December 2, 2014. 3
Mxmum Mrks: 18 Totl tme llowed: 35 mn Dvdng both sdes of the bove expresson by V yelds, V V ( ) 2 ( ) 2 ( ) 2 ly l z lx l z lx l y l x + l y + l z, l x l y l z l x l y l z l x l y l z ( lx ) 2 ( ) 2 ( ) 2 ly lz + +, l x l y l z (0.01) 2 + (0.01) 2 + (0.01) 2 0.017%, 0.02%. (2) For temperture vrtons, ll sdes re ffected eqully. Therefore, one cn use the formul for volume wth equl lengths, V l 3, nd the uncertnty n V s, V ( V ) 2 l l 3l 2 l. Dvdng by V on both sdes gves, V V 3l2 l l 3 3 0.03%. ( l l ), 3 0.01%, Ths result shows tht the overll uncertnty cn ncrese, f uncertntes re not ndependent nor rndom. 4. Fgure (3A) shows the poston x(t) versus tme plot for n elevtor cb tht s ntlly sttonry, then moves upwrd (whch we tke to be the postve drecton of x), nd then stops. Choose the best opton for the velocty v(t) nd ccelerton (t) shown n Fgure (3B). [3] Soluton: The correct nswer s (d). The slope (v dx/dt) of x(t) s zero n the ntervls from to b nd t pont d, ths mens tht the cb s sttonry. Durng the ntervl bc, the slope s constnt Dte: Tuesdy, December 2, 2014. 4
Mxmum Mrks: 18 Totl tme llowed: 35 mn x(t) c d b 0 (A) t b c b c Ι d t 0 ΙΙ t d b c d b c d 0 ΙΙΙ 0 ΙV (B) FIG. 3: (A) Poston versus tme grph, (B) grphs for velocty nd ccelerton. v(t) vs t (t) vs t () II IV (b) I II (c) I IV (d) II III (e) I III nd nonzero nd the cb moves wth constnt velocty ndcted by bc n subfgure (II). Snce the cb ntlly begns to move nd then lter slows to stop, v vres s ndcted by the slopes of b 0 nd c 0 n the subfgure (II). Thus, subfgure (II) s the requred plot. The ccelerton of prtcle t ny nstnt s the rte t whch ts velocty s chngng t tht nstnt. Grphclly, the ccelerton t ny pont s the slope of the curve of Dte: Tuesdy, December 2, 2014. 5
Mxmum Mrks: 18 Totl tme llowed: 35 mn v(t) t tht pont, therefore, dv dt d ( ) dx d2 x dt dt dt. 2 Comprng subfgure (III) wth subfgure (II), ech pont on subfgure (III) shows the dervtve (slope) of the v(t) curve t the correspondng tme. When v s constnt, the dervtve s zero nd so lso s the ccelerton. When the cb frst begns to move, the v(t) curve hs postve dervtve (the slope s postve), whch mens tht (t) s postve. When the cb slows to stop, the dervtve nd slope of the v(t) curve s negtve, hence (t) s negtve. 5. The perod of osclltons T of body constrned to rotte bout horzontl xs for smll mpltudes s gven by the expresson, ( ) 1/2 I T 2π, (1) mgd where m s mss of the body, d s the dstnce between center of mss (CM) nd the xs of rotton nd I s the moment of nert (MI) bout the xs of rotton gven by (from prllel xs theorem: I I o + md 2 ). Here I o s the moment of nert bout prllel xs through center of mss. If k s the rdus of gyrton tht depends on geometry.e., k l2 + b 2 12, then I 0 mk 2. Now Equton (1) cn be wrtten s, k T 2π 2 + d 2. (2) gd How would you fnd the vlue of g by plottng? [3] () T versus d. (b) T 2 d versus d 2. (c) T 2 versus d. (d) T versus log(d). (e) All of the bove. Soluton: The correct nswer s (b). Dte: Tuesdy, December 2, 2014. 6
Mxmum Mrks: 18 Totl tme llowed: 35 mn By lookng t Equton (2), one cn tell tht f we plot T versus d tht wll follow prbol trend. The best wy s to lnerze the gven functon whch s n mportnt technque from dt nlyss perspectve. Rerrngng Equton (2) yelds, ( 4π T 2 2 d )d 2 + 4π2 k 2. g g Ths s strght lne functon wth T 2 d s the dependent vrble, d 2 s the ndependent vrble, (4π 2 /g) s the slope whle (4π 2 k 2 /g) s the ntercept. The g vlue cn be computed through the resultnt outcome of slope (4π 2 /g). 6. Lssjous fgures re used for the mesurement of phse nd produced when one sgnl s connected to the vertcl trce of the osclloscope nd the other to the horzontl trce. If the two sgnls hve the sme frequency, then the lssjous fgure wll ssume the shpe of n ellpse. The ellpse s shpe vres ccordng the phse dfference between two sgnls nd ccordng to the rto of mpltudes of the two sgnls. The phse dfference cn be clculted through the followng expresson, Y H x 0.2V (b) () 2ms FIG. 4: () The output sgnl of n osclloscope, nd (b) n ellpse. sn(ϕ) ± Y H, (3) where H s hlf the mxmum heght nd Y s the ntercept on the y xs s shown n Fgure (4b). Dte: Tuesdy, December 2, 2014. 7
Mxmum Mrks: 18 Totl tme llowed: 35 mn Clculte the phse dfference ϕ nd ts uncertnty (ssume ths s n nlog scle) bsed on the Lssjous pttern gven n Fgure (4). [3] d (Hnt: dx (sn 1 u) 1 du ). 1 u 2 dx Soluton: Snce ech block on the osclloscope screen s equvlent to 0.2 V, therefore by redng the scle, the vlues of the ellpse s prmeters becomes, Y 0.2 + 0.05 0.25 V, X 0.4 + 0.04 0.44 V. Substtutng these vlues n Equton (3) yelds, ( ) Y ϕ sn 1, H ( ) 0.25 sn 1 34.62, 0.44 35. Snce ths s n nlog scle, uncertntes ssocted wth Y nd H cn be found out s, The vlues of Y nd H cn be quoted s, u Y 6 0.04/2 6 0.008 V, u H 6 0.04/2 6 0.008 V. Y (0.250 ± 0.008) V, H (0.440 ± 0.008) V. Notce tht uncertnty hs only one sgnfcnt fgure nd the decml plces of both the orgnl quntty nd the uncertnty re t the sme poston. The uncertnty n the phse ϕ cn be clculted through the Tylor seres pproxmton, ϕ ( ) 2 ( ) 2 ϕ ϕ Y Y + H H. (4) Dte: Tuesdy, December 2, 2014. 8
Mxmum Mrks: 18 Totl tme llowed: 35 mn Dfferenttng equton (3) w.r.t Y gves, ( ) ϕ Y 1 d(y/h) 1 ( ), Y 2 dy H 2 ( ) 1 1 1 ( ), Y 2 H H 2 1 1 ( ) 0.25 2 0.44 2 2.76. Dfferenttng equton (3) w.r.t H yelds, ϕ H 1 1 ( ) Y 2 H 2 ( ) 1, 0.44 ( d(y/h) dh 1 1 ( ) ( YH ), Y 2 2 H 2 ( 1 0.25 1 ( ) 0.25 2 0.44 2 0.44 2 1.57. Substtutng n Equton (4) results n, ), ), ϕ (2.76 0.008) 2 + (1.57 0.008) 2, 0.025 0.02. The uncertnty vlue n degrees would be, ϕ 0.02 ( 180 3.14 ) 1.1. Hence, the fnl vlue of phse ϕ cn be quoted s, ϕ (35 ± 1). Dte: Tuesdy, December 2, 2014. 9
Mxmum Mrks: 18 Totl tme llowed: 35 mn Formul sheet: Tylor seres pproxmton: If quntty q q(x, y, z) s mesured usng some nput vrbles x, y nd z whch re mesured wth uncertntes x, y nd z, respectvely, then q cn lso be fnd out usng the Tylor seres pproxmton gven s, q ( ) 2 ( ) 2 ( ) 2 q q q x x + y y + z z. Stndrd devton: s d2 N. Stndrd uncertnty: σ N (s). N 1 Stndrd uncertnty n the men: σ m σ N. Weghted verge: x vg w x w Slope (m) nd ntercept (c) wth equl weghts: m y (x x) (x or m N x) 2 N x y x y x 2 ( x ) 2 (5) c ȳ m x or c Uncertnty n slope m nd ntercept c s gven s, where, u m u c N d 2 x 2 y x x y N x 2 ( x ) 2. (6) D(N 2), (7) ( )( 1 N ) N + x2 d 2, (8) D (N 2) d y mx c, N D (x x) 2. Slope m nd ntercept c wth unequl weghts Dte: Tuesdy, December 2, 2014. 10
Mxmum Mrks: 18 Totl tme llowed: 35 mn The weghts re recprocl squres of the totl uncertnty (u Totl ), w 1. (9) u 2 Totl The mthemtcl reltonshps for slope (m) nd ntercept (c) re, m Σ w Σ w (x y ) Σ (w x ) Σ (w y ) Σ w Σ (wx 2 ) (Σ w x ) 2, (10) c Σ (w x 2 ) Σ (w y ) Σ (w x ) Σ (w x y ) Σ w Σ (w x 2 ) (Σ w x ) 2, (11) where x s the ndependent vrble, y s the dependent vrble nd w s the weght. The expressons for the uncertntes n m nd c re, Σ w u m Σ w Σ (w x 2 ) (Σ w x ), (12) 2 u c Σ (w x 2 ) Σ w Σ (w x 2 ) (Σ w x ). (13) 2 Dte: Tuesdy, December 2, 2014. 11