Physics Lab 8 copy

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Physics

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Jan 9, 2024

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Laboratory Manual for Physics 27SO 2022 " 23 SIMPLE HARMONIC MOTION LAB: DATA AND ANALYSIS SECTION A spring is suspended vertically with a mass m hanging from its lower end. The spring is stretched and the mass m is in equilibrium. 1. In the box at right, draw a free body diagram for the mass m. Clearly show and label all the forces acting on it. 2. Write the equilibrium equation for mass m. This is equation 1. ~ Q,,- VYl8 :O 3. Write the expression for the elastic force in the spring using Hooke's law. This is equation 2. 4. Draw a qualitative graph of the elastic force F. vs stretch, Lix, of the spring. What does the slope of this graph represent? \t1 // 'I - - --~ I X 5. Using equations 1 and 2 find the elastic constant of the spring as a function of the mass m, the stretch~. and any other constants. This is equation 3. Select one of the two springs provided and hook the spring to the ruler provided. Hold the ruler vertically such that the spring hangs freely. 6. Hang different masses to the free end of the spring (you will need to use at least 5-7 different masses). Make sure that you do not stretch the spring too much and destroy it (that means you fail the lab!). Increase the hanging masses slowly, step by step. 7. Using the ruler provided measure the extension of the spring from the equilibrium (with no hanging mass). 8. Record both mass and spring extension in the table below . Calculate elastic constant k using equation 3. 9. Repeat the measurements with the second spring . 71
Laboratory Manual for Physi cs 2750 2022-23 Mass (kg) Stretch (cm) k1 (N/cm) Mass (kg) Stretch (cm) k2 (N/cm) o. os 0. 5 0 - C\ i, C).C)_s I . -b D. :,06q S63 () . 07 ' . \ 6 0-lf -'\ \ °1~=> D-07 Z .C\ b. -z. ~6 ?C\3 o,oq 0-2.C\4-3 ().09 4- o. Z. 7-o7Z.S 0. \ I ' 4--4 - o. 1.. 4-'oZ. t?) O · \ \ S-3> D . · -z_(P;;, 6c4 0, \1 ~ -7 0, 1- Z-'b737 0,\3 6. 4- o. \ qq 2-bb 0-\ S 7 o. 2-\0Z..\A- o. \S 7. &" D. \nbs4 D. \ 7 0 -~ D,1fffi2-- '6 D. \7 q 0 . l YS3 7. Using LoggerPro, plot the Force acting on the spring vs Stretch for both springs on the same graph. Fit a line through the data points for each graph. Attach the graph with the fitted lines at the end of your lab report . 8. Calculate the slope of each graph. Include units in your answers. l( , ·. 0. 1# 6 Nt c VV1 \(2-: D. \G03 \\1/cV'V\ 9. Compare the values obtained from the slope of the graphs with values from the table (that you calculated). What does slope of F vs stretch represent? Is there a difference? Why or why not? -\ Y\ .Q,, S \ 0 P D \? \J s ~\-~ '( 'e.,. ,_ C,, v' l t> , (? { ' Q. "::> 1 S t Vl £- ':> p f'i vi 9 c, D V1 S -i a.,/'. - :- 10. Next, you will analyze the motion of a mass hanging from a spring. Consider a spring of length L that is connected to a support. A small mass m is connected to the free end of the spring. The mass is pulled down and the spring stretches a distance A from its equilibrium position. When the mass is let go, it starts oscillating up and down . The position of the mass on the spring changes with time as shown below. 72 ,,,. ,,,,,. ,,,. ,,,,, ,,,. ,,,, j j
Laboratory Manual for Physics 2750 2022-23 °""''"IA~ 'f~Am.n -- ~ ---- -1- ---- 1- --~--- 1 I I I - r I - - - r - - 7 ---- T ---- r - - - - r - - r -- 7- -r- -- ~ ---- -1- ---- 1- --- -- ~- -- 1- --- , __ _ ---- ' ---- ' ---- , ____ , ---- ' ---- , ____ , ____ 1 ____ ' ---- ' ________ _! t 11. What type of motion does the above graph describe? Explain your answer. ') i V'\ uSOl o\Ct I mo i · ,0v1 of CV1 0 loJ e ct LVI Pf ev1e,u ti v:!_9 cu r le,, .._ bot i om c,f -e,uc-v, 1AJCc \J e. , \I Q.Ac?L1 -\-~ i~ lt-\i9~s.t cu f (.,,Q, Q.,q ui Ii ()V'i U Wf '{t)01 v1i 12. On the graph, identify the amplitude and period of motion for the oscillating mass . Also identify and label the points of maximum and zero velocity, and maximum and zero acceleration for the oscillating mass . 13. How is the angular frequency in SHM connected to the mass and elastic constant of the spring? Wr i te the mathematical equation connecting these physical quantities. This is equation 1. W ;J~ 14 . From equation 1, find the elastic constant k. This is equation 2. 1, 1 z - ~ - \'\.'\ 15. Sketch a qualitative graph of ro 2 as a function of 1/mass. What does the slope of this graph represent? Make sure your set up is as described under Directions. Follow those instructions to collect data. 16 . First use the spring with the smaller elastic constant (from the previous activity) 17. Click on the position vs time graph and then on "Autoscale". Make sure the graph you obtained is a smooth sinusoidal curve. If the curve has jagged parts, repeat the experiment until you get a smooth curve. 18. Fit a curve to the position vs time graph. Write below the equation of the curve you fitted to the data. X -. 0 . 0 \ 4 l 6 1 i vt ( l 7, 6 C\ t 1- ") . 7 lf O ) 1- C) . DO I D 4- 7 19. What is the value of the angular frequency ro of the motion (from the fitted curve)? I 7. 6q ,/' o. .dd s 20. Print the graph with the fitted curve and append it to your lab report. Save this run as you will be using it in the next activity too. 73
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La boratory Manual for Physics 2750 2022-23 21. Repeat the experiment with 4-6 different masses and record the angular frequency from the fitted graph in the table below. Do not print the graphs. 22. Calculate the elastic constant k using equation 2 and record its value in the table below. Mass (kg) Angular Frequer\cy (rad/s) ,. . Elastic Constant ( ) 0,05 \ 7.69 \S.6 4- G ~O=> 0,07 I ZJA9 \ r t.\ ~%07 o.oq \2. . t-1- l4- - /'1 ot Q . \ \ \ \.G\ S l 1; . 7 03Z- 7S 0 ,\?, l \ .oi \s. 7 87 201 Q. \.S \0, 2.b ts .740 \4- 23 . Using LoggerPro, plot ol as a function of 1/mass. Fit a line through the data points. Print out the graph with the fitted line and add it at the end ofthe lab report. 24. What is the slope of the ro 2 vs 1/mass graph? Include the units. l S. q b f:~ Js 2 25. Compare the value for the elastic constant obtained from slope with the value from the table (that you calculated). What is the relative error between the two results? , ·, I'.: t \ \ S ? c1 - \ ts . --~ o I ma-ss c;,..,+ CJ . \S ~J .---- ,------- -- ~ :1 {?O l S . c,i6 -:: \ .Ob';'& 26. Where does the difference come from? i t 9 ro b euo \ \J co VV) e s -0 (' v 1'°'/1 VI' ,(' C)r/1 ::, , W hQ,,vt cec\ CU Lu -+ I Yl c., t S f> I' ! /"I__C{ CO rl\ t (j_r,1.J' S . J J l \ }'\, e__ UV1 Q.\JQ..- Vl ~c?v'C.e- 27. Compare the value for the elastic constant obtained from the slope with the value obtained in the previous activity where the hanging mass was stationary. What is the percentage difference between the two results? K_ z. ~CM t-J/ l'Yl I 10 . OJ -I S,qb \ (J O - 4-f d/ \ S . Cl. (;, X l - , to 28. Where does the difference come from? 'i 1 \ q ::_ Q., \ '-I V\C{.,$ t O do { ,v ii V\, tLL V0<2.-' I VlO d 1 -s \- 9 C,(.,,V' t w Ci S et s 1 01., t- , cw1 a V1 l/ vh DL s. S av, of Z-111 cA u) U S, C\.,Vl OS ( i {( u. f-i tt'"t} VV1 a S S 74
Laboratory Manual for Physics 2750 2022-23 29. Which method is faster? Which method gives you the best precision? Justify your answers using your data and measurements. c) S ~ QH 1c ' fA l ll·co fo s t-e.r' beruu~Q_ ad\Ju,yJce :::,e.n50 r' t e, VI 'f V l -Q. C LQ.cA , ~k-\~ 1 cc\ 2 v..JCl~ vvio~ fJ ~e c, .5Q., lo~ co.usa__ ·f ~ \Cf[Q.vl i cAitf'lV'e,"}CQ_ WU 0 \Jl,J/y lo w , 30 . Look at the position vs time graph you obtained. Find the average amplitude for the SHM of the hanging mass. Consider 4 -5 oscillations only. Record the average value for the amplitude below. o. 0 \4-\b VV7 31. Calculate the total energy of the oscillating system. Write the equation you used and show your calculations below. ·\ " A 1- :. b i (_ ( Cj . C{6) [_ 0 . Ot4-lb) 2 -=- o. 0016~ 32 . In LoggerPro, go to Data->New Calculated Column. Name the column Elastic Potential Energy (EPE) and include units. In the expression, enter the equation for the elastic potential energy (kx 2 /2) as follows: (the value for the elastic constant that you have found previously) * (select the column for y position using Variables (Columns)) * (select the column for y position using Variables (Columns))/ 2. Click OK. Describe how the elastic potential energy of the system changes with time and why. I l_Q_,, +: t IS VJ I t)' w :, t CLf- t- (Q,,, \o o t f-o n-, of- f- lQ. () -s GI l \ CL -\- , o vi 'Z:> ,- V'l c .e... t- n; 5 i :s w vi, +- lQ__ ~p ('1 V\Cj I':} VM c?'S t (, c'.)yV),,p /1 ~5SQ.d 33 . Print the graph for the elastic potential energy (don't forget to label it) and on the graph, indicate for two different times when the oscillating mass has maximum velocity, has zero velocity, has maximum acceleration, and has zero acceleration. 34. In LoggerPro, go to Data->New Calculated Column. Name the column Kinetic Energy, short name KE . Don't forget to include units. In the expression, enter the equation for the kinetic energy (mv 2 /2). Use the velocity column to calculate kinetic energy . Describe how the kinetic energy of the system changes with ti me and why. \LL \c.. L c}I\ CA,V) Cj,,lt")> t-o ~ of ..e..-vi i ,c.vl (A.<:, t \c, :J {) I/ ·1 \/ \~ Q..Q.._ Co ~ S ()n COW/ pr . a. ~sed t t i v, c,1 1 e(). <;JLS 75
Laboratory Manual for Physics 2750 2022-23 35. Print the graph for the kinetic energy (don't forget to label It) and on the graph, indicate for two different times, when the oscillating mass is at a maximum distance from the equilibrium, passes through the equilibrium position, has maximum acceleration and has zero acceleration. 36. Insert a New Calculated Column for the Total Energy. Total energy is kinetic energy plus elastic potential energy. Add the two columns for kinetic and elastic potential energy to find the total energy of the system. Describe the graph you obtained for the total energy of the system. fs it constant? Explain. iAfCt l\'-( 1t- ~ hu u \e,\ t) e,,,, ff t,XL lU cl in5J Cl-1V" v"e:::> \ ;st UY-le £, s l'VI()\, \ l -V"1 t' C? v"S co.. u ~CA t-- o t u I .R.Y\fl/f/ 'i ( r' -e o1) to Vl o +-- vicA\l't i v'u~ e.q v , Ii lo f'i vvvz 37. Does the value for the total energy of the system obtained from the graph agree with the one you calculated in question 36? Explain why or why not. l,lJ , _,-, -e..,lA/, vi 5' . 't-'~ \J cd v Q,5, olu i/h ce, i c l-- 1 b o t vt vi u~ vVlvC{J' vi , t u o,(e _ v a \ v e f- L l 0t...,'/--- ()Vl'V2- e. k: f- r-~l Lf b v'V1 cc,{ I 76
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Labora tory Manual for Physics 2750 2022-23 Conclusions: The conclusions should include the following parts: a) What is the purpose/goal of this lab? b) Summarize important results, methods used to obtain them, and draw conclusions supported by your results. c) Specify one or two interesting things that you learned from this lab . C\.) The; l? Li'' '.PD~'<-. o~ t'vl\S \ CL.lo i 'c:> \o cl~~e,,t/'m 1~ th,ll.. Q>.CX -. J 1 £ (_Q(/\'z>\0-V\\- C>\- CL ~ Qf'r~ U ~lVl <J -\-wo VV1 e..1l-1ovl ~ . b) Tl.,Q,- r-i v1 S t- YVl Q,,-t lt-tecR., vVe, m -e c cs uv-ilcA. et CS ()vii(l,~ vno vi~ \,\.) i 1-vi Vl(_Jf-Q..f" o,,v, o\ ,ouv-iel s pf"1n3 co n5 f-ceP1-f ..s f r" on-r tlc., CS tv1e-tcl-1 . T Vl t lQ.., '::> .fl.co viol Vv\Q..,,t vt.vd, u.,,,.e..,, lJ 0., ~ -~J,,1,S, C) V\ i O i V) nQ_ v" d) ffi-+- t-~ cwlft U { C,l,V' fVle,c(u ency Dt Cx.. fJ V'r V9 vY'\ 0 v, vi 9 _ J,- v1 c,0V1c l u0rcJ"'1, rkQ, saoV1cl te0-\- LUQs vvic:>v"'e... ac.cur-a:le__. c) fo u vi c,(_ , i- r V\ te-vi e,~ 1 , v1&' vi u vU ot, ~-«~vi '2YL f- -e, q_ u i (0 ~i- CWl.,CA vne..,'{ viocts. ~VlDvU oil t-~i f- Y' Q..,~u I ts,_ 77
Data Set 0 0 Stretch Fo rce cm N 1 0.5 0 4905 • - 2- 1.16 0 6867 - 3- 3 0~829 1.5 - 4 - 4.4 1 0791 5.7 1.2753 7 1 4715 Li near Flt for : Data Set I Force 8.3 1 6677 y = mx•b m (Slope) 0.1446 Nian D (Y-ln tercep t} : 0.4580 N Corre l ation :0.9973 RMSE: 0.03391 N 15 7'6 17 18 7'9 20 21 QJ 22 23 u. 24 25 26 2"T 28 29 0.5 30 ""'31 32 33 34 35 36' 37 38 39 40 41 42 0 43 44 0 2 4 6 8 Streich (c m)
Data Set D Stretch Force (cm) (N 1 1.6 0 4905 • 2 2.9 0 6867 3 4 0 8829 1.5 4 5.3 1 0791 5 6.4 1 2753 6 7.8 1 4715 Linear Flt for: Data Set I Force 7 9 1 6677 y =m:ic•b 8 m(Slope) 0. 1603 N/ cm b (Y-lnlercept ): 0.2330 N CorrelaUon: 0. 9995 RMSE ; 0. 0124 7 N f OJ LL 0.5 [ 0 0 2 4 6 8 Stretch (cm)
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Latest D Posibon Velocity s m mis 0. 05 -0 013 0.130 ,. 2 0.10 --0008 0163 0. 01 0.15 0003 QJ67 0.20 0.013 0.084 '\ 0. 25 0015 -0 055 \ 0. 30 0 006 -0 .155 Auto Flt for: Latest I Position 0. 35 -0006 -0. 141 g X = A' ! ln(BI.C )- 0 \ A 0.01416•/·6. <0<E-O0S \ 0.40 -0 013 -0 024 C B: 17.69 +/. 0.01685 0 0.45 -0009 0.110 0. 00 C: 3. 780 +/ . 0.01023 0.50 0.002 0.161 0 D: 0.001047 +/- 4.649E-005 0.094 Q_ 0. 55 0.012 Correlation : 0.9900 0. 60 00 15 -0 042 RMSE : 0.0002052 m \ 0. 65 0 007 -0 148 0. 70 -0 005 -0 147 0. 75 -0 013 -0038 -0. 01 0. 80 -0 010 0.100 ., 0.85 0000 0.164 ] 09 _0 0.012 0.117 I I 0. 95 00 15 0 013 00 0.5 1.0 1.00 0008 -0 .074 (0 1331, 0.013195) nme (s) 10 + 5 I 2:- u 0 J 0 Position 0.008 m -5 o '. 5 1'.o 0.0 nme (s)
Data Set -{] 1/mass omega•2 L (kg) (rad/s) 4- 1 20 3129 4 • 300 2 14.29 2399 4 3 11 .1 1 164 86 4 9. 09 14 28 5 7.69 121 44 6.66 10527 Linear Fil fo r: Data Set I omega-'2 y =m x+b m( Slope ) 15 . 96radlsA<g b (Y;nteia,pt): -1 .890 rael/S Correlation :0. 99.C9 RMSE: 8. 963 rael/S 200 i Jg_ N Q) E 0 100 0 0 20 40 60 80 100 1/mass (kg)
20 t: 0 C 0 -Fllorl _ l _ l • 4•ttn!9t<}-() AO 057• 9 •I- 0 000)91) 8 10211 •l-000<907 C Jtl1• ~ 00 1' 20 D -0 ~t:JS •I- 0 0002tO I COff - 0"11 RUSE 0001790m 1 1 /\1 ._ ., vc.l o, : ,( 1/ l c r# " ccf ., .,.-, ; _,, f'1 ; ,, vd• ,: -1-, / fl1,_ a,uf-. ,.,,., .... , _Jime (S)
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1 1v, 2167 21)81 0 wv 1198 0026 -1253 00"11 3630 001) .. 733 ooo, 715 0001 .J~O 0000 1~ OOH 067S 00,0 2 793 0013 '25,C 0003 '620 0000 367S 0007 17W ODl!l -0 57 00111 :, ~7S 0013 .c o:,a 0 00,I .C 480 0000 ''Mn n""" 0 10 0 05 .. -005 -0 10 T 0 M,1/1 T 1 2 T T T mt (S) I 3
Latest EPE KE TE 0.10~ ~ ----------------------------------------------------------- {J) (J) (J) 0098 0.002 0.100 ,. = __ 0. 08J__ OQ(g 0.083 0 087 0.008 0. 095 0.05 0.046 002EL 0.072 ,:;;- - 0008 0.027 0.036 - llQOQ. f--- 0.013 0.013 0.001 0.002 0.003 - 0.001 0. 001 O.,QQ2_ w I- 2 w 0. 00 "" 0.000 0.009 0.009 0 005 0017 0.022 0 023 0.020 0.043 0.052 0.013 0065 0 077 0.003 0.081 w n. w I -0.05 " 0087 0.000 0.087 0067 0.007 0 074 - 0038 0.016 0.054 0. 013 0019 0.032 -0.10 000 1 0.013 0.014 - 0 0~~ ~004 0.004 000 1 0.000 0. 00 1 n "'"' I"\'"'~ n nn~ 0 2 3 4 Time (s)