Force and Acceleration on an Airtrack

Transcription

1 Force nd Accelertion on n Airtrck Objectives: Experimentl objective Students will verify Newton s second lw of motion. Lerning objectives (students should lern ) The significnce nd use of Newton s second lw of motion To interpret physicl mening from grphs Equipment list: Airtrck (trck, shuttle, spring/pulley ssembly, blower), string, slotted weights, 5g mss hnger, 2 photogtes w/ stnds, computer interfce Apprtus: Photogte 1 Photogte 2 Shuttle String Spring Pulley Mss Hnger Airtrck Theory: Newton s 2 nd lw of motion Newton s Lws of motion hve been long stnding stndrd for exmining mechnics in clssicl situtions. The second lw in prticulr is of gret use nd importnce. You probbly recognize this lw in the form F=m (force is the product of mss times ccelertion), but Newton himself stted A chnge in motion is proportionl to the motive force impressed nd tkes plce long the stright line in which tht force is impressed (Philosophie Nturlis Principi Mthemtic). Mthemticlly, this lw is generlly expressed s force being the time derivtive of momentum. But then where does this F=m come from? To nswer tht, we strt with the momentum formultion, F = dp dt P t Where P is momentum nd t is time. We cn then substitute mss times velocity for momentum (from P=mv). F = d(mv) dt (mv) t Then, if we ssume tht mss does not chnge in this sitution, we cn seprte the mss term. F = m dv v m dt t

2 And you should be fmilir by now tht the time derivtive (or chnge in velocity over time intervl) is equivlent to ccelertion, which gives the fmilir F=m. It is wrrnts sying, though, tht one should be little creful with this eqution it is only vlid for specil cses. In the steps bove, there re two mjor problems, ssumptions tht re not lwys vlid. First, momentum is not lwys mss times velocity. For exmple, chrged prticle moving through n electric field, or objects in reltivistic situtions both hve different definitions of momentum. But in clssicl mechnics, p=mv is correct; this is the sme s everydy events tht you cn imgine hppening round you. Second, mss is not lwys unchnged. Imgine rocket engine. It is propelled forwrd by burning the fuel nd forcing the exhust out of nozzle t very high speed, which cuses ccelertion in the opposite direction. In this cse, s the fuel is burned nd converted to exhust, the rocket loses mss. Tht sid, F=m is fine to use s long s the two types of situtions bove re not fctor (it works gret for the scope of this course), but if one of those two situtions comes into the picture (most often it is the chnging mss), new pproch hs to be tken. In SI units, mss is expressed in kilogrms [Kg], ccelertion in meters per second, squred [m/s 2 ], nd force is in Newtons [N], which is equivlent to [Kg*m/s 2 ]. Theoreticl ccelertion in this experiment, you will ttempt to verify Newton s second lw by using known force to ccelerte frictionless object. You will then compre the ctul ccelertion of tht object to the theoreticlly predicted ccelertion. A mss hnging from string over pulley will ct s the ccelerting mss, pplying force to ccelerte the object being observed. The force this ccelerting mss (m) pplies cn be determined by multiplying the mss by the ccelertion of grvity (g=9.81m/s 2 ), F=mg. The ccelerting force is then responsible for ccelerting not only the object in question, but the ccelerting mss s well (the two msses comprise single system for this prt of the clcultion). The ccelertion, then, cn be clculted from the eqution, Or substituting for F, = F = m + m m g m + m (1) Where m is the ccelerting mss nd m is the shuttle (or object ) mss. The reson both msses (glider nd hnger) pper in the denomintor is becuse both objects re ccelerting due to the pplied force of grvity, nd thus both must be ccounted for in the mss term of F=m. Experimentl ccelertion While the theoreticl ccelertion is clculted from Newton s second lw, the experimentl ccelertion is clculted from the fmilir =Δv/Δt, which mkes discussion of the experimentl timing scheme crucil. There re three regions of time mesurement. The first time, t1, is the time it tkes for the shuttle to pss through the first photogte, beginning when the front of the shuttle blocks the bem nd stopping when the bck of the shuttle clers the bem. t2 is similr mesurement; it is the time it tkes for the shuttle to pss through the second photogte. The finl time, t3, is the time between the other two; it begins when the bck of the shuttle clers the first photogte nd stops when the shuttle enters the second photogte. It is esier to conceptulize the process if we imgine replcing the shuttle with point mss hlfwy

3 between the front nd bck ends of the shuttle. Figure 1 shows the loction regions of this point mss during ech time intervl. Shuttle center line t1 t3 t2 Figure 1: experimentl timing scheme depicted s regions of loction of point on the center line of the shuttle. The verge velocity of the shuttle pssing through ech photogte cn be clculted by dividing the length, L, of the shuttle by the time it tkes to pss through the photogte, giving v 1 = L t 1 nd v 2 = L t 2 (2) Becuse the shuttle is ccelerting, the clculted velocity is the verge velocity, not the instntneous velocity. But using our imgined point mss, its instntneous velocity is pproximtely the sme s the shuttle s verge velocity. So the ccelertion cn be clculted s the chnge in instntneous velocity of the prticle s it psses ech photogte divided by the time between ech position. The problem though, is there isn t mesured time tht corresponds to the distnce between the photogtes (see figure 1). We cn pproximte it by using t3, nd dding hlf of t1 nd hlf of t2. Thus, the experimentl ccelertion is clculted with the following eqution. = v 2 v 1 t 3 + t t 3 2 (3) Note: It is importnt tht you convert your mesurements to SI units before using ny of these equtions so tht your results hve the proper units. Procedure: You will perform this experiment in two different modes: constnt totl mss, where the totl mss of the system (hnger, shuttle, nd the mss dded to ech) remins constnt, but is moved from the shuttle to the hnger between runs; nd constnt ccelerting mss, where the hnger nd its dded mss remins constnt, but the mss dded to the shuttle vries (totl mss of the system vries in this prt). Initil set-up 1. Set up the irtrck s shown in the pprtus section, with the photogtes pproximtely 50cm prt. Switch on the ir pump (blck cylinder on the floor). Check tht the trck is level the unloded shuttle, without the string ttched, should remin sttionry when plced on the trck. 2. Plug the first photogte (photogte 1) into chnnel 1 nd photogte 2 into chnnel 2 on the computer interfce. Check tht the shuttle does not hit the photogtes s it psses through, nd the photogtes trigger only on the blck shuttle (they should not be triggered by the string or the silver clip tht connects the string to the shuttle). 3. Mesure the length of the glider nd record this vlue s L. Clculte the distnce between the photogtes by observing the shuttle position when it triggers ech photogte (indicted by the red LED), nd record this vlue s D.

4 4. Weigh the unloded shuttle, nd record this vlue s ms. 5. Weigh the unloded hnger, nd record this vlue s mh. 6. Attch one end of the string to the clip on the shuttle nd the other end to the weight hnger, running the string over the pulley, so the hnger is hnging pst the end of the tble. 7. Open the Force_Accelertion_Airtrck file in the Physics Lb folder on the computer desktop. Prt 1 Constnt totl mss 8. Add two 5g nd four 10g msses to the shuttle. Note: the mss dded to the shuttle must lwys be configured symmetriclly (the sme mount on ech side of the shuttle). The hnger (5g) hs no dded mss for the first run. 9. Record the totl shuttle mss (dded mss plus ms) s m nd the totl hnger mss (dded mss plus mh) s m in the tble. 10. One student will hold the shuttle t strting position on the trck, click strt on the computer, nd then relese the shuttle (be creful not to give the shuttle push). 11. Once the shuttle hs clered photogte 2, nother student will ctch the shuttle. Be very creful to not trigger the photogtes. 12. Record the three times in your tble. Then click stop on the progrm stopping the progrm before you record the times my result in erroneous dt. 13. Reset the shuttle, nd move 10g of the mss from the shuttle to the hnger. The totl mss of the system should be the sme, but distributed differently between the shuttle nd hnger. Don t forget to keep the mss symmetric on the shuttle. 14. Repet steps 8-12 until you hve recorded dt for t lest four different mss distributions. Prt 2 Constnt ccelerting mss 15. Adjust the mss on the hnger to one of the previously tested vlues. 16. Remove ll mss from the shuttle. 17. Record m nd m. 18. Perform run like you hve done before for this mss configurtion. 19. Add mss symmetriclly to the shuttle nd repet the dt collection until you hve recorded dt for t lest five different shuttle msses (do not chnge the hnger mss). Dt nlysis Prelb: 20. For ech run, clculte the totl system mss (m+m), ccelerting force due to grvity cting on m (F=mg), nd the expected ccelertion from eq. (1) nd record these vlues. 21. For ech run, clculte v1 nd v2 from eq. (2) nd the experimentl ccelertion from eq. (3) nd record these vlues. 22. Clculte the %error for ech of your experimentl ccelertion results. 1. If you recorded the height of the hnger during run, wht would the grph of height vs. time look like (wht would be the shpe creted)? 2. If you incresed the mss on the hnger in question one, how would the shpe of the grph be ffected? 3. How could you determine the vlue of g from grph of height vs. time without clculting ny velocities? 4. Why must t1 nd t2 be included in your clcultion for the experimentl ccelertion?

5 5. It is lwys importnt to consider the resonbility of your dt nd results to ctch mjor problems s erly s possible. List the expected order of t1-3 in terms of incresing durtion. The ccelertion must fll between which vlues (regrdless of mss distribution)? Report: Constnts Shuttle length, L = Distnce between photogtes, D = Shuttle mss (unloded), ms = Hnger mss (unloded), mh = Tble 1 prt 1 m m F t1 t2 t3 v1 v2 Expected Experimentl %error m+m = Tble 2 prt 2 m m+m t1 t2 t3 v1 v2 Expected Experimentl %error m = F = Questions: 1. How well do your experimentl results conform to your expected results? Were you ble to confirm Newton s second lw of motion? Be sure to cite your results specificlly. 2. Grph the expected nd experimentl ccelertion from prt 1 s function of ccelerting force, F. Wht is the reltionship between the expected ccelertion nd F (wht is the shpe of the grph)? Does your experimentl ccelertion exhibit similr trend? 3. Grph the expected nd experimentl ccelertion from prt 2 s function of totl mss, m+m. Wht is the reltionship between the expected ccelertion nd totl mss? Does your experimentl ccelertion exhibit similr trend? 4. Wht would you hve to do to the dt to mke the grph in question 3 liner? 5. If you were to mix up your grphs, tht is, if you grphed your prt 2 dt in question 2 nd your prt 1 dt in question 3, wht would the grphs look like?