T. LEISTI PRINTED AUGUST 2009

Transcription

1 LABORATORY MANUAL FOR PHYS*1000 An Intrductin T Mechanics E. McFARLAND J. O MEARA T. LEISTI DEPARTMENT OF PHYSICS Department f Physics University f Guelph 009 PRINTED AUGUST 009

2 TABLE OF CONTENTS Page(s) Analysis f Experimental Uncertainties 3 t 5 Intrductin t Labratry Wrk 6 Labratry Experiment #1: Intrductin t the Use f 7 t 37 Mtin Sensrs and DataStudi Labratry Experiment #: Acceleratin due t Gravity 38 t 4 Labratry Experiment #3: Mment f Inertia 43 t 51 and Angular Mmentum Labratry Experiment #4: Simple Harmnic Mtin (SHM) 5 t 60 Page

3 Intrductin ANALYSIS OF EXPERIMENTAL UNCERTAINTIES In the measurement f any physical quantity, such as length, mass, psitin, etc., there is sme uncertainty. Fr example, yu might measure the length L f a husekey by using a 15-cm ruler that has markings at 1-millimetre intervals, and determine that the length is clser t 55 mm than t either 54 mm r 56 mm. Hence, yu might feel cnfident in stating that yur value is 55 mm with a pssible range f 54.5 mm t 55.5 mm. Yu culd then write that L 55 mm with an uncertainty f ± 0.5 mm. This result culd be expressed as L (55 ± 0.5) mm r as L (55.0 ± 0.5) mm Hwever, yu might feel that yu can estimate where the ends f the key lie, between the 1-millimetre markings. Yu might feel certain that the pssible range is 55.0 mm t 55.4 mm, with the value being 55. mm. Yur measurement culd then be expressed as L 55. mm with an uncertainty f ± 0. mm: L (55. ± 0.) mm The experimental uncertainties that are being discussed here are ften referred t as experimental errrs r just errrs. Hwever, this use f the wrd errr can be misunderstd. Errr in this cntext des nt mean a mistake r blunder it simply refers t the uncertainty that is unavidable in any physical measurement. Significant Digits and Errrs When perfrming mathematical calculatins such as additin and multiplicatin, yu have prbably used rules t determine hw many significant digits t write in the final result. Fr example, when multiplying and dividing quantities, yu prbably used a rule that the number f significant digits in the result shuld equal the least number f significant digits in any ne f the quantities being multiplied and divided. These rules are actually nly apprximatins t the crrect methds (i.e., the errr analysis discussed in the fllwing pages) t determine hw many digits can be written in the final result f a calculatin. Smetimes the rules that yu have used in the past give the same number f digits as the detailed errr analysis, but smetimes they d nt. Page 3

4 Hw t State Uncertainties (Errrs) The examples in the Intrductin abve illustrate hw errrs are stated. Fr example, if a length L is being measured, then the result is written as L L ± δl where δl is the errr in the measurement f L. (The symbl δ is the lwercase Greek letter delta, and hence δl is delta L. ) Since errrs are estimates, they are nrmally expressed with nly ne significant digit. Hence, if the magnitude f the acceleratin due t gravity has been measured, the result wuld nt be written as, fr example, g (9.81 ± 0.031) m/s, but rather as g (9.81 ± 0.03) m/s. The result f a measurement is usually written with the same number f decimal places as the errr. Fr example, in g (9.81 ± 0.03) m/s, the numbers 9.81 and 0.03 each have tw decimal places. If the errr had been ±0.3 m/s, which has nly ne decimal place, then the result fr g wuld be written as g (9.8 ± 0.3) m/s. With an errr f ±1 m/s, we wuld write g (10 ± 1) m/s. Cmparing Quantities Suppse that tw students perfrm an experiment t determine the maximum acceleratin f a Frd Taurus. One student s result is that a (4.6 ± 0.3) m/s, whereas the ther student determines that a (4.1 ± 0.) m/s. The first student s result has a range f 4.3 t 4.9 m/s, and the secnd student s result ranges frm 3.9 t 4.3 m/s. Since these tw results verlap (just barely) at 4.3 m/s, then the results are equal within experimental errr. If the tw results had been a (4.6 ± 0.) m/s and a (4.1 ± 0.) m/s, then the tw results d nt verlap and are nt equal within experimental errr. Smetimes students perfrm experiments t determine a quantity that is well knwn, such as the magnitude f the charge (e) n an electrn r the universal gravitatin cnstant (G). In these experiments the knwn values nrmally have very small uncertainties. Fr example, as f July 004 the accepted value 1 f e is e ( ± ) C If a student were t measure the value f e as (1.65 ± 0.03) C, then the student s value des nt agree with the knwn value within experimental errr. Hwever, if the student s result were (1.65 ± 0.05) C, then there is agreement within experimental errr. 1 If yu are interested in finding current values and uncertainties fr varius cnstants, they are available at Page 4

5 Adding and Subtracting Quantities with Errrs The next several sectins discuss with hw t deal with errrs when using measured quantities in varius types f calculatins invlving additin, subtractin, multiplicatin, etc. Suppse that the lengths f tw different bjects are measured, and that the sum f the lengths is t be determined. The lengths f the tw individual bjects are L1 (3. ± 0.3) cm and L (51. ± 0.5) cm The range f pssible lengths fr L1 is.9 cm t 3.5 cm, and the range fr L is 50.7 cm t 51.7 cm. Hence, yu might expect that the minimum pssible value fr L1 + L wuld be ( ) cm 73.6 cm, and that the maximum value wuld be ( ) cm 75. cm. The sum L1 + L culd then be written as LSUM (74.4 ± 0.8) cm. Hwever, if the errrs in L1 and L are randm and independent f each ther, a detailed statistical analysis (beynd the level f a first-year curse) shws that the abve calculatin verestimates the errr in the sum. In ther wrds, it is verly pessimistic t assume that that the lwest value f L1 wuld add t the lwest value f L (and similarly fr the largest values). A full statistical treatment shws that the apprpriate value f the errr in the sum is given by δ L + SUM ( δl1 ) ( δl ) Thus, the errr in LSUM in the abve example is δl δ SUM L SUM ( δl ) 1 (0.3 cm) 0.6 cm + ( δl ) + (0.5 cm) Hence the sum is (74.4 ± 0.6) cm. Ntice that the errr f 0.6 cm is less than the 0.8 cm calculated using the pessimistic apprach. Adding errrs by taking the square rt f the sum f the squares f errrs is called adding errrs in quadrature. If ne quantity is subtracted frm anther, the errr in the result is als determined by adding errrs in quadrature. Fr example, using the lengths L (51. ± 0.5) cm and L1 (3. ± 0.3) cm as befre, the difference is LDIFF L L1 (51. 3.) cm 8.0 cm. The errr in this difference is δl δ DIFF L DIFF ( δl ) 1 (0.3 cm) 0.6 cm + ( δl ) + (0.5 cm) Thus, the difference is (8.0 ± 0.6) cm. Page 5

6 If many quantities are being added and/r subtracted, then the errr in the final result is just the additin (in quadrature) f the errrs f all the quantities. Summary: Suppse that a quantity q is determined as a result f additin f several quantities a, b, c,, and subtractin f several quantities x, y, z, : The errr (uncertainty) in q is given by q a + b + c + x y z δ q ( δa) + ( δb) + ( δc) ( δx) + ( δy) + ( δz) +... Fractinal Errr and Percentage Errr The errrs discussed s far, such as the ±0.6 cm in (8.0 ± 0.6) cm, are referred t as abslute errrs. Here are the results f tw measurements with the same abslute errr: (100 ± 1) cm, and (10 ± 1) cm. Althugh the abslute errrs are identical, fr the measured value f 100 cm the errr represents nly 1/100 f the measured value, whereas fr the 10- cm value the errr is 1/10 f the measured value, and is thus, relatively speaking, a much larger errr. In ther wrds, a given abslute errr can be cnsidered large r small depending n the size f the measured value. T take bth the size f the errr and the measured value int accunt, fractinal errr is ften used. If a quantity q is measured with an uncertainty δq, the fractinal errr is δq fractinal errr q (The abslute value f q is used in the denminatr t ensure that the fractinal errr is always a psitive quantity.) As an example, cnsider a measurement f length L that gives The fractinal errr is L (100 ± 1) cm δl L 1cm 100 cm 0.01 Because the fractinal errr is usually a number much less than 1, it is almst always usually multiplied by 100% t give percentage errr. Percentage errr is just a different way f expressing fractinal errr. Fr the abve example, the percentage errr is Page 6

7 δl L 1cm 100 cm % 1% Using percentage errr, we write L 100 cm ± 1%. Ntice that the abslute errr δl has the same units as L (i.e., centimetres), whereas the percentage (r fractinal errr) is a dimensinless quantity. Multiplying and Dividing Quantities with Errrs As will be illustrated in this sectin, fractinal and percentage errrs are mst useful when measured quantities are multiplied and divided. Fr example, suppse that the area A f a rectangle is t be determined using measured values f length L and width w. The area A Lw. Start with the standard frm fr the measured value f L: L L ± δl If L is factred ut n the right-hand side, this equatin can be rewritten in terms f the fractinal errr: δl L L 1 ± L Similarly, the measured value f the width w can be written as w 1 ± δw w w Since L and w are the estimates fr L and w, then the estimate f the area A will be the prduct Lw. The largest pssible value f the area is given by the prduct f the largest pssible length and the largest pssible width: largest pssible area L L δl 1 + L w w δl 1 + L δw w δw w The smallest pssible area is given by a similar equatin with the tw + signs replaced by signs. Page 7

8 Nw multiply the expressins in the parentheses in the abve equatin: largest pssible area L L w w δl L δl L δw 1 + w + δw w + δl L δw w The fractinal errrs δl / L and δw / w are nrmally small (perhaps a few percent), and s their prduct (the final term in the abve equatin) is extremely small and can be neglected. Hence, largest pssible area L w 1 + δl L + δw w The smallest pssible area is given by a similar equatin with the tw + signs replaced by signs. Thus, the area A, including its errr, can be written as A L w δl 1 ± L + δw w Cmpare this equatin with the general frm invlving fractinal area: A 1 ± δa A A Yu can see that A Lw, which we knew already, but mre imprtantly, the fractinal errr in A is just the sum f the fractinal errrs in L and w, that is, δa A δl L + δw w This result shws that if tw quantities are multiplied, the fractinal errr in the prduct equals the sum f the fractinal errrs in the tw quantities. If tw quantities are divided, rather than multiplied, it can be shwn that the fractinal errr in the result is again just the sum f the fractinal errrs in the tw quantities. Hwever, as was the case fr errrs when quantities are added r subtracted, the abve analysis is verly pessimistic. In ther wrds, it is nt highly likely that the length and width will bth have their largest values, r bth have their smallest values. A detailed statistical analysis shws that if the errrs in L and w are randm and independent f each ther, then the mst apprpriate fractinal errr in the area A is the sum in quadrature f the fractinal errrs in L and w: Page 8

9 δa A δl L δw + w where A, L,and w have been replaced by A, L, and w fr simplicity. This additin f fractinal errrs in quadrature applies when any number f quantities are being multiplied and divided, as lng as the errrs are randm and independent. Thus, we have as a general result that when measured quantities are multiplied r divided, the fractinal (r percentage) errr in the result is the sum in quadrature f the fractinal (percentage) errrs in the multiplied and divided quantities. Summary: Suppse that a quantity q is determined as a result f multiplicatin f several quantities a, b, c,, and divisin by several quantities x, y, z, : q a b c... x y z... The fractinal (r percentage) errr in q is given by δq q δa a δb + b δc + c δx x δy + y δz + z +... The three examples presented belw illustrate the cncepts develped s far. Example 1 The magnitude f the (linear) mmentum (p) f an bject is given by p mv, where m and v are the bject s mass and speed, respectively. A baseball having mass m (0.150 ± 0.004) kg is thrwn with a speed v (44 ± 1) m/s. Determine: (a) (b) (c) (d) the magnitude f the baseball s mmentum the percentage errr in the mmentum the abslute errr in the mmentum. Write p ± δp, and p ± % errr. Slutin (a) The magnitude f the baseball s mmentum is p mv ( kg)( 44 m/s) 6.6 kg m/s Page 9

10 (b) The percentage (r fractinal) errr in p is the sum in quadrature f the percentage errrs in m and v. First determine the percentage errrs in m and v: δm m δv v Then, the percentage errr in p is just kg kg % 1m/s 44 m/s 0.0 % δp p δm m 4% ( 3% ) + ( % ) δv + v (c) Frm part (b), the percentage errr is: δp p 4% Re-arrange t slve fr the abslute errr δp: δp 4% p 4% (6.6 kg m/s) 0.3 kg m/s (d) Writing p with its abslute errr: p ± δp (6.6 ± 0.3) kg m/s Writing p with its percentage errr: 6.6 kg m/s ± 4% Page 10

11 Example Slutin An autmbile f mass m is being lifted n a hist. During a time t the aut is lifted a vertical height h (at cnstant speed). If m (1.50 ± 0.01) 10 3 kg, g 9.80 m/s (assume negligible errr), h (.00 ± 0.01) m, and t (.50 ± 0.0) s, determine (a) the pwer P required t lift the aut (b) the percentage errr in P (c) the abslute errr in P. (d) Write P ± δp, and P ± % errr. (a) Pwer is energy divided by time, and since the aut is being lifted, the apprpriate energy t cnsider is the gravitatinal ptential energy, mgh. Thus, the pwer is P mgh t 3 ( kg)( 9.80 m/s )(.00 m) W.50 s (b) The percentage errr in P is the sum in quadrature f the percentage errrs in the quantities being multiplied and divided: m, g, h, and t. We are tld t assume negligible errr in g, and hence we have δp P δm m δh + h δt + t First calculate the percentage errrs in m, h, and t: δm m kg % kg δh h 0.01m.00 m % δt t 0.0 s %.50 s Page 11

12 The percentage errr in P can nw be calculated: δp P δm m 1% δh + h ( 0.7% ) + ( 0.5% ) + ( 0.8% ) δt + t Thus, the percentage errr in P is 1%. (c) Frm part (b), the percentage errr is: δp 1% P Re-arrange t slve fr the abslute errr δp: δp 1% P 1% ( W (d) When writing P ± δp, the quantity δp shuld be expressed with the same pwer f 10 as P. Since P W and δp 1 10 W, rewrite δp as W. Then, 4 W) 4 P ± δp W ± (1.18 ± 0.01) 10 W Finally, writing P with its percentage errr: W ± 1%. 4 W Example 3 Slutin If an bject underges cnstant acceleratin a, its velcity v at time t is given by v v0 + at where v0 is the velcity at time t 0. A mtrcycle accelerates frm (15.0 ± 0.5) m/s t (34. ± 0.7) m/s in a time f (3. ± 0.1) s. Determine the acceleratin (assumed cnstant) during this time interval, alng with its abslute percentage errrs. Since we are required t determine acceleratin, first re-arrange the equatin t slve fr acceleratin, and substitute knwn values: Page 1

13 v v0 a t (34 15) m/s 3. s 5.9 m/s Nw fr the errr calculatin. The abve equatin fr acceleratin invlves a subtractin, which will require additin f abslute errrs, and a divisin, which needs additin f fractinal r percentage errrs. Deal with the subtractin first. The abslute errr in v v0 is the sum in quadrature f the abslute errrs in v and v0: δ ( v v ) ( δv) + ( δv ) 0 ( 0.5) + ( 0.7) 0.9 m/s 0 m/s The quantity (v v0) is divided by t, and s the percentage errrs in (v v0) and t need t be added in quadrature. Determine these percentage errrs: ( v v ) ( v v ) ( 34-15) δ m/s % m/s 0 δt t 0.1s 3. s % The percentage errr in the acceleratin a is then δa a 6% ( v v ) δ v v ( 5% ) + ( 3% ) 0 0 δt + t Thus, the acceleratin is 5.9 m/s ± 6%. The abslute errr is 6% 5.9 m/s 0.4 m/s, and hence the acceleratin can be written as (5.9 ± 0.4) m/s. Page 13

14 Special Case: Exact Number Times Measured Quantity A special case that ccurs frequently in calculatins is the multiplicatin f an exact number (i.e., ne that has zer uncertainty) and a measured quantity that has uncertainty. An example is the calculatin f the circumference C f a circle by multiplying the cnstant π and the measured diameter d f the circle (C πd). As shwn belw, the determinatin f the abslute and percentage errrs in the result f this type f calculatin turns ut t be very straightfrward. Suppse that a quantity q is the prduct f an exact number B and a measured quantity x: q Bx Since this calculatin invlves a multiplicatin, the fractinal (percentage) errr in q is given by δq q δb B δx + x Since B is an exact number, δb 0. Therefre, the abve equatin gives δ q δx q x In wrds, this result states that when an exact quantity multiplies a measured quantity, then the percentage errr in the result is equal t the percentage errr in the riginal measured quantity. Nw fr the abslute errr in the result. Start with the abve expressin fr the percentage errr: δ q δx q x Substitute q Bx in the denminatr f the left-hand side: δq δx Bx x Cancelling the x s, and re-arranging t slve fr δq: δq Bδx This result indicates that when an exact quantity multiplies a measured quantity, then the abslute errr in the result is equal t the (abslute value f the) cnstant times the abslute errr in the riginal measured quantity. Page 14

15 Example 4 Slutin Six identical rectangular pieces f steel, when stacked n tp f each ther, have a ttal thickness f (9.0 ± 0.06) cm. What is the thickness f each piece f steel? T determine the thickness f ne piece f steel, all that is required is a divisin f the ttal thickness by 6 (an exact number). This divisin can be cnsidered t be a multiplicatin by 1/6. As shwn n the previus page, when an exact quantity multiplies a measured quantity, the abslute errr in the result equals the exact quantity (i.e., 1/6) times the abslute errr in the measured quantity. Therefre, the thickness f ne piece f steel is: 1 (9.0 ± 0.06) cm (1.53 ± 0.01) cm 6 Ntice that the percentage errr in the riginal measurement is 0.06 cm 100% 0.7% 9.0 cm and the percentage errr in the final result is 0.01cm 100% 0.7% 1.53 cm that is, the percentage errrs are the same, as stated n the previus page. Errr in a Functin f One Variable Suppse we have a functin f(x) f a single variable x, which has a measured value f x ± δx. The crrespnding value f the functin will be f(x), but what will be the uncertainty (errr) in this value? This errr δf can be evaluated as the difference between f(x + δx) and f(x): δf f(x + δx) f(x) A fundamental apprximatin in calculus states that fr any functin f and a very small increment u, Page 15

16 f ( x + u) f ( x) Therefre, if δx is small (which it usually is), the difference δf f(x + δx) f(x) can be written as df dx u f ( x + δx) f ( x r df ) δx dx df δf δx dx Since δf is nrmally a psitive quantity, and since df/dx culd be psitive r negative then an abslute value sign is needed arund df/dx, as shwn in the summary bx belw. Summary: If f is a functin f ne variable x, and if x is measured with an uncertainty (errr) δx, then the crrespnding errr δf is given by df δ f δx dx Errr in a Pwer A cmmn type f functin that arises in physical systems is a pwer functin, that is, f x n. Using the relatinship between δf and δx given in the bx abve, δf df δx dx d dx nx n x δx n 1 δx If bth sides f this equatin are divided by f x n, δf f nx x nx n 1 n 1 δx n x δx δx Page 16

17 Ntice in this equatin that δf / f is the fractinal errr in f, and δx / x is the fractinal errr in x. The abve result shws that if f x n, then the fractinal (r percentage) errr in f is n times the fractinal errr in x. Summary: If f x n, and if x is measured with an errr δx, then the fractinal (r percentage) errr in f is n times the fractinal (percentage) errr in x: δ f δx n f x Example 5 Slutin The radius f a circular cver fr a telescpe is measured t be (31.86 ± 0.03) cm. What is the area (including the percentage and abslute errrs) f the cver? The area A f a circle f radius r is A πr, where π A πr π ( cm) 3189 cm Since A invlves the multiplicatin f π and r, the percentage errr in A is the sum in quadrature f the percentage errrs in π and r : δa A δπ π δ + r ( r ) Since π is an exact number, it has n errr: δπ 0 π The percentage errr in r is times the percentage errr in r: Page 17

18 δ r ( r ) δr r 0.03 cm 31.86cm % Therefre, the percentage errr in A is δa A δπ π () 0 + ( 0.% ) 0.% δ + r ( r ) The abslute errr in A is δa 0.% 3189 cm 6 cm Hence, the area is 3189 cm ± 0.%, r (3189 ± 6) cm. Why nt the fllwing way? Here s anther way t apprach the abve errr calculatin. It gives a different (and incrrect) answer. The explanatin f why the methd is incrrect is given after the calculatin. Cnsider the calculatin f the area as A πr r, that is, as the prduct f three quantities. The percentage errr in A is then the sum in quadrature f the percentage errrs f the three quantities: δa A δπ π δr + r δr + r δπ π δr + r The percentage errrs n the right-hand side f the abve equatin are δπ 0 π Page 18

19 δr r 0.03 cm cm % Then the percentage errr in A becmes δa A δπ π () 0 + ( 0.09% ) 0.1% δr + r This result is nt the same as the 0.% calculated earlier. Why the difference? Lking back at the first calculatin, we had δa A δπ π δ + r ( r ) and δ r ( r ) δr r Substituting δr/r fr δ(r )/r gives δa A δπ π δr + r Cmpare this with the crrespnding equatin frm the secnd methd: δa A δπ π δr + r Yu can see that the appears in different places. In the first equatin, the is squared, and in the secnd ne it is nt. Hence, the first (crrect) methd gives a larger result. S why is the secnd methd incrrect? In writing the area as A πr r and calculating the errr in A as the sum in quadrature f the errrs f the three quantities, we are assuming that the errrs in the three quantities are randm and independent f each ther. (This cnditin was stated when adding errrs in quadrature was intrduced earlier.) Hwever, the errrs in the tw quantities r and Page 19

20 r are nt independent they are the same errrs. Therefre, the secnd methd cannt be used. In cnclusin, errrs in pwers (such as r ) cannt be treated as errrs in a prduct (r r). Example 6 Slutin A glfball is drpped frm rest and falls a vertical distance f (1.500 ± 0.003) m. At the lcatin where the ball is drpped, the magnitude f the acceleratin due t gravity has been measured t be (9.79 ± 0.01) m/s. Hw lng des it take fr the ball t fall? Express the errr in this time bth as a percentage errr and an abslute errr. Chse the +y directin t be dwnward. The equatin that relates psitin y, acceleratin a, and time t is y y0 + v0t + ½at Since the ball is drpped frm rest, the initial velcity v0 0, and it is cnvenient t chse the initial psitin y0 t be zer. Thus, the equatin simplifies t y ½at Re-arrange t slve fr t and substitute numerical values: t y a ( ) m 9.79 m/s s T determine the errr in t, first nte that the square rt can be written as a pwer: y t a 1/ Therefre, the percentage errr in t is simply ½ the percentage errr in y/a: δt t y 1 δ a y a Page 0

21 Nw determine the percentage errr in y/a. Since y/a invlves a multiplicatin and divisin, the percentage errr is just the sum in quadrature f the percentage errrs in, y, and a, which we calculate first. The has n errr, and the percentage errrs in y and a are: δy y m m % The percentage errr in y/a is then δa a 0.01 m/s 9.79 m/s % y δ a y a δy y ( 0.% ) + ( 0.1% ) 0.% δa + a As already stated, the percentage errr in t is ½ the percentage errr in y/a: δt t Therefre, the abslute errr in t is 1 ( 0.% ) 0.1% δt 0.1% s s Hence, the time fr the ball t fall is ( ± ) s r s ± 0.1%. Example 7 An angle is measured t be θ (31 ± 1)º. What are sinθ and the abslute errr in the value f sinθ? Page 1

22 Slutin It is easy t determine that sin(31º) 0.5. T determine the errr in sinθ, we use the general relatin develped earlier fr any functin: df δ f δx dx In the present example, x θ, and f(θ) sinθ. Thus, ( θ ) δ sin ( θ ) δθ d sin dθ The derivative f the sine functin is the csine functin: ( sinθ ) d dθ csθ Therefre, fr the errr δ(sinθ) that we are required t find: δ ( sin θ ) csθ δθ Nw, csθ cs(31º) 0.86, and we must multiply this by the errr δθ. Hwever, δθ must be expressed in radians. (The derivative f sinθ is csθ nly if θ is in radians; this unit enters thrugh the dθ in the derivative.) There are π radians in 180º, and s δθ (which is given as 1º) can easily be cnverted t radians: Therefre, the errr in sinθ is π rad δθ rad 180 ( sinθ ) In cnclusin, sin(31 ± 1)º 0.5 ± 0.0. δ csθ δθ (0.86)(0.0) 0.0 Page

23 Exercises and Prblems (answers at end) 1. Write the fllwing measured values and errrs with the crrect number f digits: (a) (6.78 ± 0.44) m/s (b) (351 ± 87) kg (c) (19.7 ± ) m (d) ( ± ) J. Cnvert the fllwing abslute errrs t percentage errrs: (a) (815 ± 9) W (b) (47.3 ± 0.6) kg m/s (c) (5.73 ± 0.04) 10 3 C 3. Cnvert the fllwing percentage errrs t abslute errrs: (a) 9.78 m/s ± 0.8% (b) m/s ± % 4. In each case belw, determine whether the tw quantities are equal within experimental errr. (a) (4.78 ± 0.04) 10 3 km and (4.76 ± 0.03) 10 6 m (b) kg m /s ± % and (1.68 ± 0.01) 10 4 kg m /s (c) 6.8 m/s ± % and 6.51 m/s ± 1% 5. Determine the sum L l1 + l + l3, as well as the abslute and percentage errrs in L, given l1 (1.7 ± 0.04) cm, l (.68 ± 0.05) cm, and l3 (0.01 ± ) cm. 6. Using the values f l1 and l frm the previus questin, determine the difference d l1 l, and the abslute and percentage errrs in d. 7. Pwer equals energy divided by time. Given a pwer f (687 ± 8) W, and a time f 15 s ± 3%, determine the energy (and its abslute and percentage errrs). 8. An elevatr is travelling upward at cnstant speed v. At a certain time the elevatr is at a height h1 (8.67 ± 0.05) m abve the grund flr f the building, and at a time f (.57 ± 0.05) s later, its height abve the grund flr is h (17.67 ± 0.07) m. Determine the elevatr s speed (and its abslute and percentage errrs). 9. Determine the kinetic energy (½ mv ) f an autmbile f mass m (1.87 ± 0.03) 10 3 kg, travelling at speed v 5.7 m/s ± %. 10. Any bject (including yu) that has an abslute temperature T abve 0 K emits electrmagnetic radiatin. Fr a very ht bject such as the sun (T 6000 K), this radiatin is visible. Fr yu (surface T 300 K), the emitted radiatin is largely infrared. The intensity I f emitted radiatin is given by I T 4, where σ (the Stefan-Bltzmann cnstant) is σ ( ± ) 10 8 W m K 4 Determine the intensity f radiatin emitted frm an bject having a temperature f (300.0 ± 0.6) K. Include the percentage and abslute errrs in the intensity. Page 3

24 11. The vlume V f a sphere f radius r is given by V 4πr 3 /3, where π Determine the radius (including abslute and percentage errrs) f a sphere having a vlume f (1.000 ± 0.001) m An angle φ is measured t be (8.0 ± 0.7)º. Determine csφ, and the abslute errr in this value. 13. As shwn in the diagram, tw sides f a right-angle triangle have been h (0.18 ± 0.007) m measured. (a) Determine angle θ. θ (The errr in θ will be determined in the fllwing l (0.3 ± 0.008) m parts.) (b) Use tan θ h/l and the given errrs in h and l t determine the percentage errr in tanθ. (c) Nw that [δ(tan θ)]/tanθ is knwn frm part (b), determine the abslute errr in tanθ, that is, determine δ(tanθ). (d) Knwing that d(tanθ)/dθ sec θ, determine the abslute errr δθ in angle θ. (e) Write yur final answer fr θ alng with its abslute errr δθ (with the angles in degrees). Answers t Exercises and Prblems 1. (a) (6.8 ± 0.4) m/s (b) (3.5 ± 0.3) 10 3 kg (c) (0 ± ) m (d) (6.73 ± 0.0) 10 4 J. (a) 815 W ± 1% (b) 47.3 kg m/s ± 1% (c) C ± 0.7% 3. (a) (9.78 ± 0.08) m/s (b) (.73 ± 0.05) 10 8 m/s 4. (a) equal (b) nt equal (c) nt equal 5. L (4.41 ± 0.06 cm) 4.41 cm ± 1% 6. d ( 0.96 ± 0.06 cm) 0.96 cm ± 6% 7. energy J ± 3% (8.6 ± 0.3) 10 4 J 8. v 3.50 m/s ± % (3.50 ± 0.07) m/s 9. kinetic energy J ± 4% (6. ± 0.) 10 5 J 10. I 459 W/m ± 0.8% (459 ± 4) W/m Page 4

25 11. r m ± 0.03% (0.604 ± 0.000) m 1. csφ ± (a) 39º (b) 6% (c) 0.05 (d) 0.03 rad (e) θ (39 ± )º Page 5

26 Intrductin t Labratry Wrk The primary purpse f the labratry experiments is t give yu sme direct experience with the phenmena discussed in lectures. In additin, physics experiments teach techniques f measurement and data analysis that are valuable in any labratry r measurement situatin. It is hped that these experiments will help yu t develp the skills necessary fr accurate labratry wrk. Preparatin Befre cming t the lab, read the experiment yu are t perfrm. It is nt necessary t understand the experimental prcedure in detail, but yu shuld have a general idea f the physical prperties invlved and the experimental apprach t be fllwed. Recrding Data The result f every measurement shuld be recrded in a permanent recrd as each measurement is made, either thrugh the data acquisitin sftware yu will use in the experiments r with a pen r pencil in the spaces prvided in the labratry experiment utlines. One shuld never recrd experimental bservatins and results n `rugh' paper, etc., and then transfer the data later t a final "neat" reprt. This leads t destructin f the riginal data and pssible transcriptin errrs. In the interest f neatness and cnciseness, it is helpful t recrd data in clumns in tables whenever apprpriate. Be sure t recrd all units and t recrd the data t the crrect number f significant digits. Include experimental uncertainties ( errrs ) where indicated. The instructins fr each experiment will typically guide yu t the apprpriate frmat fr data recrding. Analysis After the data are recrded, make the calculatins and plt the graphs indicated in the utline. Again, shw all units and recrd the crrect number f significant digits. Perfrm errr analysis if required. The lab reprts required in this curse are nt elabrate. The reprt is simply yur data, recrded in tabular r graphical frm, as well as calculatins (including errr analysis) and discussin as utlined in the experimental instructins. The reprt will be checked t see that yu have dne the experiment cmpletely, with reasnable care and accuracy, recrded the crrect number f significant digits and have prper units fr all quantities. Yur discussin f the results will als be examined t determine yur level f understanding f the physical principles being explred in the experiment. The experiment and reprt are t be cmpleted in the labratry perid. Page 6

27 Objectives LABORATORY EXPERIMENT #1 INTRODUCTION TO THE USE OF MOTION SENSORS AND DATASTUDIO At the cnclusin f this experiment, the student shuld: be able t use efficiently the data acquisitin sftware (DataStudi) assciated with the sensrs t be used in subsequent experiments have a better understanding f the physical interpretatin f psitin vs. time and velcity vs. time graphs have gained sme experience in dealing with experimental uncertaintities ( errrs ). Discussin The experiments in this curse will require data cllectin using mtin sensrs r rtatinal mtin sensrs and their assciated sftware, called DataStudi. The first activity f the semester is designed t help yu t becme familiar with this apparatus, s that future experiments will be cnducted efficiently. In additin, the prcedure will ask yu t reprduce a psitin vs. time graph, as well as a velcity vs. time graph, using yur mtin sensr. These particular activities have been chsen in rder t reinfrce yur understanding f the cncepts f psitin, velcity, and acceleratin used t describe mtin in ne-dimensinal kinematics. Self Test I (answers are given n the last page f this lab utline) 1. A train mves alng a lng straight track. The graph shws the psitin as a functin f time fr this train. The graph shws that the train: a) speeds up all the time. b) slws dwn all the time. c) speeds up part f the time and slws dwn part f the time. d) mves at a cnstant velcity. Page 7

28 . A persn initially at pint P in the illustratin stays there a mment and then mves alng the axis t Q and stays there a mment. She then runs quickly t R, stays there a mment, and then strlls slwly back t P. Which f the psitin vs. time graphs belw crrectly represents this mtin?. # # # # # # EXPERIMENTAL PROCEDURE Part 1: Preparatin Open the DataStudi sftware by duble clicking n the icn n the desktp. The sftware shuld already be aware f the mtin sensr cnnected t the cmputer thrugh the USB prt. When DataStudi pens, select the Create Experiment icn, and clse Graph 1 that pens autmatically using the X in the tp right crner f the Graph 1 windw. Click k when asked if yu want t cmpletely remve this display. The screen shuld nw lk like this: Page 8

29 On the left-hand side, the tp half shws the data yu will acquire, and the bttm half shws the varius display ptins available t yu fr yur data. We will almst exclusively be using the graph display ptin in this curse. - When yu pen the sftware, the left-hand side f the screen will lk as shwn n the left here. - Nw let s cllect sme data. But first, make sure the switch n the sensr is n the wide-beam setting (the stick figure setting ( )), nt the narrw-beam setting (the cart setting). The sensr is nw ready t send ut a narrw beam f ultrasund waves, and the sftware will recrd the distance frm the sensr t a reflecting bject, like yur hand r a bk, by timing the ech. - Click n the Start buttn at the tp f the screen, mve yur hand back and frth in frnt f the sensr, then hit Stp after a few secnds. - Under the Psitin (m) subheading, there is nw a set f data called Run #1 ; let s graph this t see what it lks like. - Drag Psitin (m) dwn t the Graph icn under Displays ; the left-hand side f yur screen will nw lk as shwn n the right here. Page 9

30 - Frm a trial experiment perfrmed earlier, ur psitin vs. time data fr mving a hand away frm the sensr, and then twards the sensr, are shwn belw: - The details f yur graph will prbably lk very different. - Yur mtin sensr is als capable f displaying velcity vs. time data fr the same run. Click n Setup near the tp f the screen, and check the bx fr acquiring velcity data, then minimize the Setup windw by clicking the - buttn at the tp right. - The left-hand side f yur screen will nw have tw subheadings under Data : Psitin (m), as befre, but nw Velcity (m/s) shuld als be shwn (see belw). - Plt the velcity vs. time data frm the same experiment ( Run #1 ) by dragging Velcity (m/s) dwn under the Graph subheading belw. - Frm ur trial experiment perfrmed earlier, the velcity vs. time data are shwn belw graphically: Page 30

31 - Yu shuld be able t see that these velcity vs. time data crrespnd t the psitin vs. time data shwn in the previus pht. - Yu shuld practice acquiring and pltting data with yur sensr. Click Start again, and Run # will autmatically be displayed n the existing Graph 1. - If yu wuld like t display these data separately, drag Run #1 r Run # frm the tp half f the left hand side dwn t the Graph icn belw t create anther graphical display, but with just this ne data set. - S, remember that if yu drag an entire data subheading (like Psitin (m) ) dwn t the Graph icn, all runs under that subheading will be displayed n ne graph. But, if yu drag just ne run t the Graph icn, nly ne data set will appear n the graph. - Yu shuld get in the habit f renaming each graph, in rder t make clear the specific experiment that was perfrmed and the names f the peple invlved. Yu will be generating print-uts f these graphs, s yu want t be able t identify them afterwards. T rename a graph, under the Displays heading n the lwer left hand side, click n the graph yu want t rename, then click again nce, and type in a new name in the bx that appears. Page 31

32 - T remve data frm a display, click n the data set yu wuld like t remve in the text bx shwn in the upper right crner f the graph display, and then hit the Delete key. T remve an entire display, use the X bx in the uppermst right hand crner f the display windw. While testing ut the sensr and the sftware, investigate the fllwing: change the sample rate, using the + and - buttns, thrugh the Setup buttn near the tp f the screen set a time delay fr data cllectin (handy fr getting the experiment set up befre the recrding begins), thrugh the Optins buttn under the Setup windw set a preset length f time fr data cllectin, als thrugh the Optins buttn under Setup (see Autmatic Stp ) use the varius buttns acrss the tp f the graph windw t explre their functins: Scale t Fit Zm In Zm Out Zm Select Smart Tl Slpe Tl Fit Calculate Statistics Nte In particular, yu will find the Scale t Fit, Zm Select, Smart Tl, and Slpe Tl buttns useful in future experiments. print ut a graph, after yu have labelled it apprpriately, by pulling dwn the File menu n the tp left hand side and selecting Print. Each lab partner shuld have ne print-ut t submit t the Teaching Assistant (TA) frm these preliminary tests. Part : Matching Graphs 1. G t the file menu in DataStudi, and select Open Activity. Nte: Whenever yu are asked if yu wish t save an activity, always respnd by clicking N.. Open the file match_psitin.ds, and fllw the n-screen instructins. Yu may find it helpful t hld a bk in frnt f yu when yu try t match the graph n the screen. (The bk gives a better ech.) When yu have cmpleted this matching activity, print a cpy f yur graph with an apprpriate title. BEFORE YOU CLICK OK in the print dialg bx, make sure yu are nly printing the 5 th page f the activity (select the Pages ptin n the Print range sectin f this bx, and enter frm 5 t 5.) Make sure that each lab partner has a turn at this activity, and that each partner has a print-ut f page 5. Clse this activity. Page 3

33 3. Open the file match_velcity.ds, and fllw the n-screen instructins. When yu have cmpleted this matching activity, print a cpy f yur graph with an apprpriate title. Make sure that each lab partner has a turn at this activity, and that each partner has a print-ut. Clse this activity. 4. Fr the match_psitin.ds activity, use the space prvided belw t: - describe in wrds the mtin taking place - sketch the crrespnding velcity vs. time graph 5. Fr the match_velcity.ds activity, use the space prvided belw t: - describe in wrds the mtin taking place - sketch the crrespnding acceleratin vs. time graph Page 33

34 Part 3: Displacement, Velcity, and Errrs In this part f the experiment, yu will create a graph f psitin vs. time, and then calculate displacement and average velcity (with experimental errrs). Set the sample rate t 0 Hz. Cllect psitin data fr several secnds while mving yur hand r bk away frm the sensr at cnstant velcity (as clse t cnstant as yu can manage). Create a graph f psitin vs. time. Use the Smart Tl t determine the time and psitin values fr a pint n yur graph that ccurred early in the data cllectin. Enter the values belw as time t1 and psitin x1. Include the experimental uncertainty in the value fr psitin: ± m. (This uncertainty was prvided by the manufacturer f the apparatus.) Assume negligible errr in the time. (Remember units.) Yu and yur lab partner shuld chse different pints n the graph fr the time and psitin values. t1. x1. Use the Smart Tl t determine the time and psitin values fr a pint n yur graph that ccurred later in the data cllectin. Enter the values belw as time t and psitin x. Include the experimental uncertainty in the value fr psitin: ± m. Assume negligible errr in the time. t. x. In the space belw, calculate the displacement )x x!x1. Als calculate the experimental errr in )x by adding the errrs in x1 and x in quadrature. Express this errr as bth an abslute errr and a percentage errr. Page 34

35 In the space belw, calculate the average velcity vav between times t1 and t. Als calculate the experimental errr in vav, and express this errr bth as an abslute errr and a percentage errr. Turn ff the Smart Tl by clicking the Smart Tl buttn. Use the cursr arrw t draw a bx arund a regin f yur graph that is very clse t a straight line. Click the Fit buttn, and then click Linear Fit n the menu that appears. A bx will appear that gives the parameters fr a linear equatin that fits the graph regin that yu have selected. (The linear equatin is x mt + b [just the same as y mx + b, but with x replaced by t and y replaced by x].) The slpe m is the average velcity fr the graph regin yu selected, and the Standard deviatin m is the abslute errr in the value f m. In the space belw, enter the value f the average velcity (m), alng with its errr. Yu and yur lab partner may use the same values fr this part f the experiment. Page 35

36 In the space belw, cnvert the abslute errr in the average velcity (m) t a percentage errr. Print a cpy f the graph (ne fr yu and ne fr yur partner), apprpriately labelled and shwing the bx with the value fr m and its errr. At the end f the labratry perid, yu shuld each hand in t the TA: a print-ut f an apprpriately labelled graph frm yur preliminary tests f the sensr and sftware a print-ut f yur match_psitin.ds graph a print-ut f yur match_velcity.ds graph the crrespnding analysis and sketches (n pg. Intrlab-8) f the match_psitin.ds and match_velcity.ds activities pages Intrlab-9, 10 and 11 shwing data and calculatins a print-ut f the final graph yur answers t Self Test II (see belw) questins 1 and nly. Include with yur answer a), b) r c) an explanatin as t why this is the crrect chice. (Space is prvided fr yur explanatin fllwing each questin.) Self Test II 1. This psitin-time graph shws an example f: a) psitive acceleratin b) zer acceleratin c) negative accleratin + psitin time + Page 36

37 . This velcity-time graph shws an example f: a) negative acceleratin b) psitive variable acceleratin c) psitive cnstant acceleratin + velcity time + 3. In rder t display in the graph windw the value f the area under the data set, yu need t: a) calculate it by hand and write it n the print-ut b) put a check mark beside Area under Statistics c) use the Calculate tl 4. In rder t display in the graph windw the slpe at a particular pint in the data, yu need t: a) calculate it by hand and write it n the print-ut b) put a check mark beside Slpe under Statistics c) use the Slpe tl and mve it t the particular pint with the muse 5. In rder t display in the graph windw the values f the x and y crdinates at a particular lcatin, yu need t: a) calculate them by hand and write it n the print-ut b) put a check mark beside Crdinates under Statistics c) use the Smart Tl and mve it t a particular lcatin Answers Self Test I 1: b, : # Self Test II 3: b, 4: c, 5: c (Nte: technically yu culd answer a) fr #3, #4 and #5, but the sftware will likely give yu a mre precise calculatin.) Page 37

38 LABORATORY EXPERIMENT # ACCELERATION DUE TO GRAVITY Equipment List Cmputer with DataStudi sftware 1 air track (Fig. 1) 1 mtin sensr, munted at the raised end f the track, high enugh ver the track s that an air car can pass underneath when that end f the track is prpped up 1 air car (large) 1 metal blck (7.618 cm Η 5 cm Η 5cm) used t prp up ne end f the air track 1 metre stick Figure 1 Mtin sensr, air car, and air track befre being raised at ne end. Intrductin In this experiment yu will determine the magnitude f the acceleratin due t gravity, g, by measuring the acceleratin f an air car alng an inclined air track. Errr Analysis A cmplete errr analysis is required fr this experiment. Yu will be expected t recrd experimental errrs in all recrded data, and t carry errrs thrughut all yur calculatins t determine the final experimental uncertainty in yur value f g. Prcedure and Dataa Cllectin Befre prpping up ne end f the air track, measure the distance L between the centre f the air track s duble-ft supprt (near ne end f the air track) and the centre f the single-ft supprt (near the ther end). Yu shuld think abut the way t make this measurement t minimize the experimental errr. Recrd yur value belw. distance L ( ± ) cm. Nw place the metal blck under the single-ft supprt (but see imprtant nte belw) t elevate that end f the air track. Set the lngest dimensinn f the blck vertically (Fig. ). Imprtant nte: There are tw airtracks in the lab that are smaller than the ne pictured in Figures 1 and. On each f the smaller airtracks Page 38

39 The mtin sensr is C-clamped at the end ppsite the air intakes, similar t the large tracks. This end f each small track has tw small feet while the air-intake end has asingle feet. Fr these smalll tracks, the end with tw feet is the ne t be prpped up, and tw metal blacks are prvided fr this purpse. Open DataStudi. Set the sample rate t 0 Hz. Set the mtin sensr t recrd nly velcity, and set up a graph f velcity vs. time. Figure Using the metal blck t prp up the track. Check that the switch n the mtin sensr is set fr a wide beam (indicated by an icn f a persn ( ) beside the switch). Make sure that the sensr is aimed alng the air track. T d this, place yur head just abve the centre f the air track, and while lking at the sensr, have yur partner adjust the sensr s that yu can see the reflectin f yur eye in the mirrred frnt surface f the sensr. It is imprtant t make sure that the sensr is nt aimed t high r t lw. When everyne is ready, the TA will instruct yu t turn n the blwers. Try a few practice runs frm the lwer end f the track give the air car a quick push up the track s that it then glides almst the entire length f the track tward the sensr, but des nt get clser than abut 30 cm t the sensr. Allw the car t glide back dwn the track, but stp it just befre it reaches the bttm f the track. Yu want t cllect a graph f velcity vs. time fr the car fr the whle trip as it glides up and dwn the track. Perfrm several runs ne after anther, s that the blwers d nt need t be turned n later. (They re nisy.) Obtain 3 gd sets f data they can all be n the same graph fr nw. (Delete bad runs.) In this experiment yu and yur partner can use the same data sets. When everyne in yur area has cllected data, the blwers shuld be turned ff. T view a graph f nly ne set f data at a time, delete the graph with all the data n it, and then click-and-drag each data run separately t Graph n the Display menu t create three separate new graphs with just ne run each. Page 39

40 Data Analysis Fr each f the three data runs perfrm the fllwing steps: Use the Fit buttn t perfrm a Linear fit f the data fr abut a twsecnd segment f the graph, centred at the turning pint where the car turned arund and its velcity was zer. Recrd the slpe (i.e., the acceleratin a) and the standard deviatin f the linear fit belw. Run #1: acceleratin a1 ( ± ). Run #: acceleratin a ( ± ). Run #3: acceleratin a3 ( ± ). Fr ne set f data, give the graph an apprpriate title (include yur name), and print it. Print a separate graph with yur partner s name n it. In the space belw, calculate the average value, a, f the three acceleratins, and determine the experimental errr in a. (In mre advanced curses yu will learn a mre sphisticated methd f calculating the errr in an average, but fr nw just perfrm the errr calculatin by recgnizing that the calculatin f a invlves nly adding the three acceleratins and dividing by the exact number 3. Yu already knw hw t deal with errrs when adding and dividing numbers.) Calculatin f a (± errr): Page 40

41 The magnitude f the acceleratin f the air car is a gsinθ, where θ is the angle f incline f the air track frm the hrizntal. Yu have determined a, and nw t determine g, yu must first calculate θ. Yu have already measured the distance L between the supprts f the air track. Nw that the track has been prpped up, this distance L is alng the incline and frms the hyptenuse f the right-angled triangle shwn. The height h f the metal blck used t elevate ne end f the air track is h (7.618 ± 0.004) cm Knwing L and h, determine sinθ and the errr in sinθ in the space belw. Calculatin f sinθ (± errr): θ L h Page 41

42 Writing sinθ with its errr: sinθ ±. Nw that yu knw the values f a (± errr) and sinθ (± errr), determine g: Calculatin f g (± errr): When yur calculatins are finished, clse DataStudi and remve the metal blck that is prpping up the air track. Cnclusins and Discussin At Guelph, g has been measured t several significant digits. Runded t three digits, the value is 9.80 m/s. Des yur result agree with this value, within experimental errr? Discuss pssible ways t imprve this experiment. Page 4

43 LABORATORY EXPERIMENT #3 MOMENT OF INERTIA AND ANGULAR MOMENTUM Equipment List Cmputer with DataStudi sftware 1 rtary mtin sensr with attached metal disc and three-part pulley 1 vertical supprt rd and stand 1 string, kntted at ne end and lped at the ther 1 5-g mass hanger 1 50-g mass 1 large metal ring triple-beam balance (balances are available at varius lcatins in the lab rm) masking tape Intrductin In this experiment yu will determine the mment f inertia f a rtating system, and then test the law f cnservatin f angular mmentum. A rtary mtin sensr interfaced with a cmputer willl be used t display graphs f angular velcity vs. time. Part 1: Mment f Inertia Prcedure The apparatus is shwn in Figure 1. A metal disc is attached t a hrizntal shaft that passes thrugh the rtary mtin sensr, which is munted n a vertical supprt rd. When the disc is spinning, the shaft spins, as well as a pulley (nt visible in Fig. 1) attached t the ther end f the shaft. In this part f the experiment yu will determine the ttal mment f inertia f all the spinning cmpnents (disc, shaft and pulley) by applying a trque t this system and measuring the angular acceleratin. Open DataStudi and select Createe Experiment. Click the Setup buttn. Set the sample rate t 50 Hz. Uncheck Angular Psitin and check Angular Velcity (nte the units). Minimize the Setup Windw. Figure 1. Disc attached t rtary mtin sensr. Page 43

44 Clse the graph which is n the screen. Click Angular Velcity under Data and drag it int Graph under Displays. Make sure the sensr is securely attached t the rd. Figure shws the pulley that is attached t ne end f the shaft f the rtary mtin sensr. It cnsists f three circular discs (shwn), each with a different diameter. Hk the kntted end f the string thrugh the ntch in the side f the largest-diameter this largest-diameter disc, and then wind the string arund disc, leaving the lp at the end f the string hanging slightly free. Hld the pulley statinary and attach the 5-g mass hanger t the lp end f the string. Place the 50-g mass n the mass hanger. While still hlding the pulley (Fig. 3), click Start. Release the pulley, allwing the 55-g mass t fall. Catch the mass just befre it hits the table tp (but allwing the string t cme cmpletely ff the pulley). Click Stp. Use the muse t highlight the linear regin f the graph f angular velcity vs. time (the regin when the mass was falling). Click the Fit buttn and select Linear Fit frm the pull dwn menu. Recrd the slpe f the line and the standard deviatin Figure. Three-part pulley. Figure 3. Preparing t release the mass. (experimental uncertainty) f the slpe, which is the angular acceleratin α. Recrd α as a psitive quantity (we want nly the magnitude f, nt its directin): α ( ± ). Insert a title (including yur name) n yur graph, and print the graph (make sure that the linear-fit bx is still being displayed). Repeat the entire prcedure t btain a separate printed graph fr yur lab partner. Page 44

45 Determining the f Mment f Inertia Befre actually calculating the mment f inertia I f the spinning system, yu need t derive an algebraic expressin fr I. Draw a free-bdy diagram fr the falling mass, and a free-bdy diagram fr the spinning pulley (+ shaft + disc). Derive an expressin fr I in terms f the falling mass m, the radius r f the large part f the three-part pulley, the measured angular acceleratin α, and the magnitude f the gravitatinal acceleratin g. In yur derivatin remember that the tensin f the string is nt equal t mg, because the mass is accelerating dwnward. Space is prvided belw fr yur derivatin. (There is als a trque due t frictin acting n the spinning system, which will nt be included explicitly in yur calculatin. This trque is nt cnstant; it depends n angular velcity, and hence is difficult t evaluate numerically. The mment f inertia that yu calculate will be smewhat larger than the true mment f inertia because the effect f the frictinal trque has effectively been included in the mment f inertia.) Derivatin f expressin fr mment f inertia: Page 45

46 Calculatin f mment f inertia: Calculate I using the fllwing data and yur measured value f α. Errr analysis is nt required fr the calculatin. m (0.055 ± 0.001) kg r (0.038 ± 0.000) m g 9.80 m/s (negligible errr) Page 46

47 Part : Angular Mmentum Prcedure In this part f the experiment yu will rient the metal disc hrizntally (Fig. 4), give it a spin, and then drp the metal ring nt the spinning disc. The ttal angular mmentum f the spinning system befre and after drpping the ring will be calculated. Accrding t the law f cnservatin f angular mmentum, these tw angular mmenta shuld be equal. In the data windw delete Run #1 and Linear Fit. Lsen the screw hlding the sensr n the vertical rd and lift the sensr ff the rd. Turn the sensr by 90Ε and re-attach it firmly t the rd, with the shaft in a vertical pstin and the metal disc n tp (Fig. 4). The sensr shuld be nly a few centimetres abve the tabletp. Click the Setup buttn, and set the sample rate t 0 Hz. Minimize the setup windw. Click Start. Give the disc a light spin. Hld the metal ring 1 r cm abve the spinning disc (Fig. 5) and drp the ring s that the centre f the ring lands as clse as pssible t the centre f the disk. (On the disc there is an aiming circle f the same diameter as the ring t help yu with this step.) Click Stp. If the ring landed badly ff centre, delete Run #1, and repeat the previus fur steps. If the ring slid n the disc after it was drpped, try spinning the disc mre slwly, r putting pieces f masking tape n the disc. Once yu have had a successful run, click the Smart Tl buttn. Figure 4. Orientatin f apparatus fr Part. Figure 5. Ready t drp the disc. Page 47

48 Mve the bx ver a pint n the angular velcity graph just befre the ring was drpped and recrd the angular velcity ω0 at that pint. Assume an experimental uncertainty f ±0.05 rad/s. ω0 ( ± ). Mve the bx ver a pint n the graph just after the ring was drpped and recrd the angular velcity Τf at that pint. Again assume an experimental uncertainty f ±0.05 rad/s. ωf ( ± ). Insert a title (including yur name) n yur graph, and print the graph. Click the Smart Tl buttn t turn it ff. Delete Run #1. Repeat the drpping f the ring t btain anther set f data and a separate printed graph fr yur lab partner. Replace the sensr n the rd as it was when yu came in (with the sensr high n the rd and the shaft hrizntal). Calculating Angular Mmentum Nw the angular mmentum L befre and after the ring was drpped will be calculated. Befre the ring was drpped, the initial angular mmentum was L0 Iω0 where I is the ttal mment f inertia f the disc-shaft-pulley system (determined in Part 1) and ω0 is the angular velcity befre the disc was drpped. Errr analysis is required fr the calculatin f L0. Assume an errr f ±% in the value f I; the errr in Τ0 was already prvided. Calculatin f L0 (including experimental errr): Page 48

49 Nw the final angular mmentum Lf (after the ring was drpped) will be determined: Lf (I + Ir) ωf where I still represents the mment f inertia f the disc-shaft-pulley system, and Ir is the mment f inertia f the ring, which is given by Ir ½ mr (r + ri ) where mr is the mass f the ring, and r and ri are its uter and inner radii, respectively. Use ne f the pan balances in the lab rm t determine mr: The radii are: mr ( ± ). r (0.038 ± ) m ri (0.067 ± ) m Calculatin f Ir (n errr analysis in this step): Althugh yu have enugh infrmatin t calculate the errr in Ir, t save time we prvide yu here with a typical value f ± 0.4% fr the errr in Ir t use in the calculatin f the angular mmentum after the ring was drpped. Nw calculate the final angular mmentum Lf (I + Ir) ωf. Errr analysis is required fr the calculatin f Lf. The errrs in I, Ir, and Τf have already been prvided. Page 49

50 Calculatin f Lf (including experimental errr): Page 50

51 Cnclusins and Discussin Are the initial and final angular mmenta equal within experimental errr? Suggest ways in which this experiment culd be imprved. Page 51

52 LABORATORY EXPERIMENT #4 SIMPLE HARMONIC MOTION (SHM) Equipment List Cmputer with DataStudi sftware 1 spring hanging frm a supprt (Fig. 1) 1 mtin sensr pinted upward (Fig. 1) 1 5-g hanging mass 1 10-g mass 0-g masses Intrductin In this experiment yu will investigate the psitin, velcity, and acceleratin f a mass underging simple harmnic mtin (SHM). As well, yu will explre the relatinship between the perid f scillatin and the mass f the scillating system. Figure 3 SHM apparatus. Errr Analysis The nly errr analysis required in this experiment is a cmparisn f the theretical value f the slpe f a line with the actual measured value (in Part f the experiment.) Details are prvided later in the utline. Initial Prcedure The spring des nt have a cnstant diameter. Ensure that the wide-diameter end f the spring is at the bttm. Check that the switch n the mtin sensr is set fr a wide beam (indicated by an icn f a persn ( ) beside the switch). Place the mtin sensr facing upward, directly underneath the hanging spring. Open DataStudi. Page 5