Physics Lab 6 copy

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Laboratory Manual for Physics 2750 2022-23 ROTATIONAL DYNAMICS LAB: DATA AND ANALYSIS SECTION The setup of the experiment is as follows: an object of mass m is connected to a string that is wrapped around a wheel of radius Rand mass M, as shown in the figure below. When the mass mis released, it falls and unwraps the string from the wheel, thus the wheel rotates. You must derive an expression for the moment of inertia of the wheel by applying Newton's 2 nd law to the rotational motion of the wheel. 1. Draw all forces acting on the wheel and on the hanging mass m. Build free body diagrams for each object. Free Body Diagram: Translation Free Body Diagram: Rotation M K T iCA 2. Using the force diagram you have drawn, write Newton's 2 nd Law for the translational motion of the hanging block. This is equation 1. 3. Using the forces you have drawn, write Newton's 2 nd Law for the rotational motion of wheel. This is ..J - C>\ equation 2. 17' - -r:: r- :r:o-. I -- R, 4. Write the equation that connects the angular acceleration of the wheel to the tangential acceleration of a point on its rim. This is equation 3. -:L o--. vY\ lfj -0-) = - ~, _i 55

Laboratory Manual for Physics 2750 2022-23 5. Using the three equations above, derive an expression for the moment of inertia of the wheel as a function of m (ha_nging mass), R (radius of the wheel), g (gravitational. acceleration), and a (linear acceleration of the hanging mass). This is equation 4. ~'L- I. ::- m (g - Ov ') Using the same setup you must derive an expression for the moment of inertia of the wheel by applying the conservation of energy law to the wheel + hanging mass system. Consider that the initial state of the system is after the hanging mass started moving, therefore mass m has an initial speed. 6. Write the total energy of the system in the initial state expressing kinetic and potential energy specifically for each object. This is equation 5. 1- "'-8 YV) "Z. ti - ~\Ii \- -r I w?- "Vh 9 H- initial position H final position 7. Write the total energy of the system after mass m has fallen a distance h, expressing kinetic and potential energy specifically for each object. We will call this the final state. This is equation 6. t~: \ W\ '1 f 1- T w~i. 56

Laboratory Manual for Physics 2750 2022-23 8. Write the equation that connects the angular velocity of the wheel to the linear velocity of the hanging mass for the initial and final positions. These are equations 7 and 8. \JJ:.. t I . - - ~1. W\ \I. 'Z- .\.., T .Yi!:. h t'V$3 tt t, I - 1 7,, - r,1 t,,-: t Yh'\.1(.-2- ... 1 I$ 1- 9. Combine equations 5, 6, 7, and 8 to write the conservation of energy between· the initial and final states. This is equation 9. The only variables in your equation should be I, v1, v2, m, g, R, and H. 10. Find the moment of inertia of the wheel from equation 9. This is equation 10. T : v'\rl l'J i2.. - ~.t) "'l VV1.0 tt ~z.. ($~t._\j; 2-) 57

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Laboratory Manual for Physics 2750 2022 -23 Now let's analyze the results you obtained from the video analysis. 11. Fit a curve to the y vs t graph you obtained from video analysis. From the fitted curve, read and record the position of the falling mass at time t = 0.4 s (we call this the initial position) and at time t = 0.8 s (we call this the final position). Read these positions from the curve fitted toy vs t graph, and not from the data table. Fill out the values in the table below. 12. From the fitted curve, read the linear acceleration of the falling mass and record it in the table below. Print they vs t graph showing the fitted curve and the fit values and append it at the end of your lab report. 13. Use the Vy vs t graph to find the velocities at time t = 0.4 s and t = 0.8 s by reading their values from the fitted line to the data. You may need to slightly adjust the interval over which the fitting is done. R~ad and record the linear acceleration from the fitted line. Print the Vy vs t graph showing the fitted line and append it to the end of your lab report. 'l. A 14. Calculate the average of each quantity and record it in the last shaded row of the table . . - yi (m) yt(m) a(m/s2) v;(m/s) v1(m/s) a(m/s2) (t = 0.4 s) (t = 0.8 s) from y vs t graph (t = 0.4 s) (t = 0.8 s) from Vy vs t graph ('.) . L\i O- b15 \ .6'b 0,55b \. 2-2-7 \.67t f) .\C\O ().~~ \. 66 D.SbS \. 21>7 \. 6)9 O-Z-b7 D.o\~ \.66 0,707 · '. ~42- ,. st9 o. \i":> o.vt3 \.?t\-1- O-stt l. ZA-t l-lSO o. iq s~ 0,766 I. l b\ O.?'DS7 1.48"73 I. 7Cfr o. 2-2-C\ . o-6o q ~ -~ C\S 0-6~6 \.~ 7 - ~ -tt 15. Is there a difference between the values of the linear acceleration obtained from the position graph and the velocity graphs? Explain. \J O , 'o e,, (,C, £. A) SQ,, tl-Q. ,v ' -e.,, \ s CL O. 0 'O S e,l ) W .e,VlC ~ W\r\ ,c\r\ C. UY"\ et..CW U V\ i--er.A.. . WV' olu <l to Vl \J vv, liv vt f?)/' V--OV', 'S O ttLVl V\ I c c\., Ill/ \f Q ) 'o u\- -, t '> V\~\ 'i Ho\,Q . 58

Laboratory Manual for Physics 2750 2022-23 16. Calculate the moment of inertia for the wheel using equation 4. ! -: ¼ : ~(9) -~) ~6f O,$Ct?;\ - ~-b~) K:O.~ \ a W'\:0 - \ 17. Calculate the moi:r_ent of inertia of the wheel using equation 10. 1:. ~ l\J ; -\l~ "t..) l--2.Wi e,\-\ R 2 H:. o.~cs L ~~ 'l. -\1 ~ ) _ Q,\ to,b~e,.i.. ,\ . 1>0 7 '1.) ..-2. ·0.\ .q, i1 ·0.3K' - (1 .- '3>072-_ 0 . 6s~ -i-) J: :. 0' 4 7 2-- \,3 VY\ 18. What is the relative error between the two results? fQ;\OvTI\l{,,, Q,,fV'()V' : "- J\ - - \j , x 2.00 A,,~ ( (9.-472 - o.t:1-: ~ ~ zbo - (?. 4-7Z-+-0 ~ -f3b5 -=- 9 % 19. What are some possible sources of error for this experiment? \-\ $ VV\CU/\ if{' (} v' \ S -\-~ VVl O '5 \- Q f'C) bC"0bl Q., C,CL USQ,, of- 0v'fY.Jv' cllJ.e,,, ta \t\0vv, v'\9 to et _ c.,c.,uruf~t l, c\ 1c,'R.. 59Q,c.,, ~-, c_ 90, v\ ts t-o o lo \c,L\ vx_ -\'(Q_ ct cv-\- C\_ 59

Laboratory 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. goo.,\~\$ \0 JQ;t ev" VV\-\V\Q..., VV\ OM Bv\,r- cif - ,nQ,,v t,C\. % {' D'-- w~ \ tY\ eov" e,. :\,cct.-lly ClVlcA expe,v, ;Y\{U1f-Cd ·iy. D) l)~ 1V\e 109~ pro ~o . o b-\- cu , v1 clv'-'C(.1 tve us.tcA, n- \-(9 t -, v1 ct t lt,, V\r\C, V\Ae,, v v~ -- o {}- 1 V\ e v' t-, C\., Vo rvyJ 2-- e,,q u w--, o v\s , l/2, v\5' eq uvv\-, o Y1 4- we..- ~o u Vlol ,r- VJ Q__, c? -4-1v:) k.gW12., lJ~ "i v\9 eqva,,t,~On tO UJQ.,; U VlcA..- \ t- 10 b-e.,,. 0.4-?Z-- ,~v ,/1/1 z. 1-n COrJC) (.J'S ton, +l-v t-w o · w1Q.v"t 1Q' s WQ., ~o uvtcl a,V1 e..,. vwy v \ c?~ \,V) tlr\ C\_ er~ 0/'rof' . C) } ~ Ou V'lcL \ t- W\, '$ t-~ to L) VI clt,o CLV\ct.\ 'S I s O 11 l Ufj~ p {'(} CNV\-vl r- cPou nd \Y - ,v,, te,v",Q~ ti v1 EJ VI OL,\) Gloe)_ cfc:JSJc, 1-uL CX.Cv Q,,,~ -0V' Ol,-\- ·1 o v'lS L.ve,,v1e,... 60

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