The Atwood Mchine OBJECTIVE To derive the ening of Newton's second lw of otion s it pplies to the Atwood chine. To explin how ss iblnce cn led to the ccelertion of the syste. To deterine the ccelertion produced by n unblnced ss condition. To study how the frictionl force nd the pulley's inerti ffect the behvior of the Atwood chine. INTRODUCTION Fro Newton's second lw: When net force cts on ss, ccelertion is produced. The Atwood chine tkes dvntge of this lw to produce n ccelertion of set of sses. The Atwood chine is two sses connected to the ends of string looped over pulley. In the presence of n unblnced ss the Atwood chine produces net force which cuses n ccelertion of the syste. One ss oves upwrd s the other ss oves downwrd. Mesureents of this ccelertion cn be deterined fro tie nd distnce esures which re then copred to theoreticl predictions bsed on Newton's second lw. APPARATUS Coputer Vernier coputer interfce Logger Pro Vernier Photogte with Super Pulley Mss set String THEORY Newton's second lw is given by: F = Eqution Where, F [N] is the net force, [kg] is the ss nd [/s ] is the ccelertion. The Atwood Mchine - Pge
Let us pply Newton's second lw to the Atwood chine. We will nlyze the forces tht re present on ech of the sses in the syste. Below is representtion of the stndrd Atwood chine. T T Figure Where, [kg] is ss #, [kg] is ss # nd T [N] is the tension in the string (due to grvity cting on the sses). To best nlyze the behvior of the two sses we will picture ech ss with the forces tht ct on ech respectively. T T Figure g g In blnced condition (equilibriu) = nd the sses rein sttionry. This, however, does not llow the study of Newton's second lw. Thus, let us consider the cse where >. This condition gives greter downwrd ttrction nd thus, it ccelertes downwrd while (ttched vi light string to ) will ccelerte upwrd with the se ccelertion s. The downwrd ccelertion of will not equl the ccelertion due to grvity s the string, pulley, nd re inhibiting free-fll sitution. Therefore pplying Eqution to Figure : T = T = g + g - Eqution Where, [/s ] is the ccelertion of the syste. Eqution is derived fro the su of the forces in the y-direction. The Atwood Mchine - Pge
Becuse the string is connected between nd, nd fro Newton's third lw, the tension in the string is the se for both nd. Thus, setting the two equtions in Eqution equl bsed on the equl tensions gives: ( g + g - g - - = or g = - or g = or ) g =( Eqution 3 g - - + + Where, ( - )g [N] is the pplied force tht cuses the ccelertion [/s ]. Note, the ccelertion of the syste, bsed on Newton's second lw, is derived solely fro pir of sses nd grvity. To this point we hve ignored the friction in nd the inerti (tendency to rein t rest) of the pulley. These quntities cnnot, however, be neglected. As ss is esure of inerti, percentge of the pulley's ss cn be used s esure of the pulley's inerti (one-hlf the totl ss of the pulley). To be ore ccurte in Eqution 3, by including the frictionl forces (pririly of the pulley itself) nd the pulley's inerti, the following derivtion is used: ( - F net ) g - = F - f or =( f Eqution 4 = + ) + Where, f [N] is the frictionl force (opposite the pplied force) nd p [kg] is the ss of the pulley tht fctors into the inerti of the pulley. Both the frictionl force nd the inertil ss will be experientlly deterined quntities. p ) The Atwood Mchine - Pge 3
EXPERIMENTAL PROCEDURE Figure 4 40 c Prt I: Keeping The Totl Mss Constnt For this prt of the experient you will keep the totl ss used constnt, but ove weights fro one side to the other. The difference in sses chnges.. Set up the Atwood s chine pprtus s shown in Figure 4. Be sure the hevier ss cn ove t lest 40 c before striking the floor or tble top.. Connect the Photogte with Super Pulley to DIG/SONIC of the interfce. 3. Open the file 0 Atwoods Mchine in the _Physics with Vernier folder. A grph of velocity vs. tie will be displyed. 4. Arrnge collection of sses totling 00 g {50g hnger, 00g, 0g, 0g, 0g, 5g, 5g} on nd 00 g {50g hnger, 00g, 50g} ss on. These sses re shown in the dt tble. Position s high up s it cn go nd relese. Wht's the ccelertion of this syste? Is this expected? Record this vlue in the dt tble. 5. Move 5 g fro to. 6. Position s high up s it cn go. Click to begin dt collection. Stedy the sses so they re not swinging. Wit t lest one second nd relese the sses. Ctch the flling ss before it strikes the surfce or the other ss strikes the pulley. 7. Click the Liner Fit button to fit the line y = t + b to the dt. Slide the brckets, [ ], into plce to select the region of the grph where the velocity ws incresing t stedy rte (positive ccelertion). Record the slope vlue (ccelertion) in the dt tble. You will need one copy of this grph, with the brckets nd dt box included, for your lbortory report. You need ONE printout totl; NOT one for ech tril. Additionlly, you will NOT need second grph for the Prt II dt collection. The Atwood Mchine - Pge 4
8. Continue to ove sses fro to in 5 g increents, chnging the difference between the sses, but keeping the totl constnt. Repet Steps 6-7 for ech ss cobintion. Repet these steps through ss difference of 50g. Prt II: Keeping The Mss Difference Constnt For this prt of the experient you will keep the difference in ss between the two sides of the Atwood s chine constnt nd increse the totl ss. 9. Put 0 g on nd 00 g on ; these sses should include the ss of the hnger!! 0. Repet Steps 6 & 7 to collect dt nd deterine the ccelertion.. Add ss in 0 g increents to both sides, keeping constnt difference of 0 grs. The resulting ss for ech cobintion re shown in the dt tble. Repet Steps 6 & 7 for ech cobintion. Repet the procedure through totl ss of 380g. COVER PAGE REPORT ITEMS (To be subitted nd stpled in the order indicted below) (-5 points if this is not done properly) Lb Title Ech lb group eber s first nd lst ne printed clerly Color Group Dte DATA (worth up to 0 points) Dt tbles vilble s downlodble Excel file ONE sple Velocity vs. Tie grph fro the experientl trils tht includes the Liner Fit dt box. DATA ANALYSIS (worth up to 0 points) Experientl Procedure Grph -- Plce nlysis on the grph printout itself -- Using your Velocity vs. Tie grph, explin wht the grph is indicting bout the otion of the sses nd why the region selected ws pproprite. The Atwood Mchine - Pge 5
Now...Reove the pulley fro the ssebly & exine it without the string nd sses hnging fro it. Give it spin with your finger. Bsed on this exintion, wht pproxitions {lrge enough to tter = you hve to keep the ter; sll enough to be considered negligible = cn set the ter equl to zero} cn you ke regrding the ss of the pulley (p; just the spinning disk prt) nd the frictionl forces (f) cting on the pulley s they relte to the experient nd the dt collected? You ll use these conclusions fro the questions bove when discussing the equtions for ech prt below. We will begin with the eqution: Prt I: Keeping The Totl Mss Constnt ( + +p ) ( - ) g - Rewrite the eqution to reflect the new p & f pproxitions you concluded. Wht quntity in the eqution would you consider to be the ss difference,? Mke this substitution nd rewrite the eqution gin. Solve this new eqution for "." It now represents liner eqution tht plots the ccelertion [y-xis] vs. [x-xis] o Like This: = (Slope Stuff) * + Y-Intercept Stuff. Your "slope stuff" y not literlly be ultiplied by. The point is tht the "slope stuff" is ANYTHING in tht prt of the eqution tht is NOT! Additionlly, the "y-intercept stuff" ight be zero. Bsed on the bsic for of the liner eqution fro the previous step, which quntities in your new eqution should correspond to: o The slope? o The y-intercept? f The Atwood Mchine - Pge 6
Prt II: Keeping The Mss Difference Constnt We will gin begin with the eqution: ( + +p ) ( - ) g - f Rewrite the bove in the se wy you did previously to reflect the pproxitions you concluded bout p & f fro your pulley exintion. Wht quntity in the eqution would you expect to be the totl ss, MT? Mke this substitution nd rewrite the eqution gin. Solve this new eqution for "." It now represents liner eqution tht plots the ccelertion [y-xis] vs. /MT [x-xis]. o Like This: = (Slope Stuff) * + Y-Intercept Stuff M T Your "slope stuff" y not literlly be ultiplied by /MT. The point is tht the "slope stuff" is ANYTHING in tht prt of the eqution tht is NOT /MT! Additionlly, the "y-intercept stuff" ight be zero. Bsed on the bsic for of the liner eqution fro the previous step, which quntities in your new eqution correspond to: o The slope? o The y-intercept? GRAPHS (worth up to 0 points) Prt I: Keeping The Totl Mss Constnt Unplug the Logger Pro fro the coputer nd open new, blnk dt tble/grph within the progr [New Pge Button/Icon]. Plot grph of ccelertion [y-xis] vs. [x-xis], using your Prt I dt. Before you print this grph, AUTOSCALE the dt nd click the Liner Fit button to fit the line y = x + b to the dt; displying the dt box contining the slope nd y-intercept dt. Prt II: Keeping The Mss Difference Constnt Agin, with the Logger Pro unplugged fro the coputer, open new, blnk dt tble/grph within the progr [New Pge Button/Icon]. Plot grph of ccelertion [yxis] vs. /MT [x-xis], using the Prt II dt. Before you print this grph, AUTOSCALE the dt nd click the Liner Fit button to fit the line y = x + b to the dt; displying the dt box contining the slope nd y-intercept dt. The Atwood Mchine - Pge 7
GRAPH ANALYSIS (worth up to 0 points) Prt I: Keeping The Totl Mss Constnt -- Plce nlysis on the grph printout itself -- Bsed on your nlysis of the grph, wht is the reltionship between the ss difference nd the ccelertion of n Atwood s chine? How do your previous definitions of the slope nd the y-intercept copre to the results tht you found experientlly; i.e. wht should be the vlue of the slope (fro the eqution) you found in the dt nlysis section vs. the vlue you ctully deterined (fro the grph)? How do you ccount for ny differences? Prt II: Keeping The Mss Difference Constnt -- Plce nlysis on the grph printout itself -- Bsed on your nlysis of the grph, wht is the reltionship between totl ss nd the ccelertion of n Atwood s chine? How do your previous definitions of the slope nd the y-intercept copre to the results tht you found experientlly; i.e. wht should be the vlue of the slope (fro the eqution) you found in the dt nlysis section vs. the vlue you ctully deterined (fro the grph)? How do you ccount for ny differences? CONCLUSION (worth up to 0 points) See the Physics Lbortory Report Expecttions docuent for detiled infortion relted to ech of the four questions indicted below.. Wht ws the lb designed to show?. Wht were your results? 3. How do the results support (or not support) wht the lb ws supposed to show? 4. Wht re soe resons tht the results were not perfect? QUESTIONS (worth up to 0 points) DO NOT forget to include the nswers to ny questions tht were sked within the experientl procedure ) How should the tension in the string chnge fro tril to tril for the Keeping The Totl Mss Constnt cses? ) How should the tension in the string chnge fro tril to tril for the Keeping The Mss Difference Constnt cses? 3) Should the ss of the string be dded to the equtions you grphed (fro pge 6) to obtin better ccurcy? Explin. The Atwood Mchine - Pge 8