Pre-Lab 4 Tesio & Newto s Third Law Refereces This lab cocers the properties of forces eerted by strigs or cables, called tesio forces, ad the use of Newto s third law to aalyze forces. Physics 2: Tipler & Mosca, Physics for Scietists ad Egieers, 6 th editio, Vol. (Red book). Chapter 4, especially sectios 4-6 (eample 4-8) ad 4-7 (also 4-8). Physics 4: Walker, Physics, 4 th editio, Vol. (Blue book). Chapters 5 & 6, maily sectios 5-4, 5-5, 6-2, 6-3 ad 6-4. Especially study Eamples 5-4 ad 6-5.. Review of cocepts Wheever you work out a problem ivolvig forces, you should draw a free body diagram. You may assume that the acceleratio of gravity g = 9.8 m/s 2, that there is o frictio or air resistace, ad ay strig is of egligible mass. Questio A block of mass m =.2 kg rests o top of aother block of mass m 2 = 3.3kg, ad that block rests o a table. a. For each block, draw a free body diagram. Idetify the source of each force i each free body diagram. b. Which of the forces that you drew are equal i magitude because of Newto s secod law? c. Which of the forces that you drew are equal i magitude because of Newto s third law? d. What is the magitude ad directio of the force eerted by the table o the bottom block? Questio 2 Two freely rollig carts, A ad B, are pushed alog a track, with cart A beig pushed horizotally by a perso s had, ad cart B beig pushed by cart A. Cart A has mass m A = 0.75 kg ad cart B has mass m B =.kg. The acceleratio of the carts is a = 0.2 m/s 2. a. For each cart, draw a free body diagram. Idetify the source of each force i each diagram. b. Which of the forces are equal because of Newto s Third law? c. Which of the forces are equal because of Newto s Secod law? d. Which of the forces are ot equal because of Newto s Secod law? e. What is the magitude of the force from cart B o cart A? 2009 Departmet of Physics, Uiversity of Washigto /6
Questio 3 Pre-Lab 4: Tesio & Newto s Third Law A freely-rollig cart rests o a iclied track that makes a agle θ with respect to the horizotal. The cart is held i place by a light strig passig over a pulley. The other ed of the strig is attached to a force sesor fied to the table, as show i the diagram. The cart has mass m =.7 kg. a. Draw a free body diagram of the cart. Idetify the source of each force i the diagram. b. For each of the forces i the diagrams, idetify the force that is its third-law pair. c. If the force sesor has a maimum measureable force of 6.0 N, what is the steepest agle that θ ca be before the force sesor is saturated? 2. Apparatus This eperimet uses the carts, force sesors ad DataStudio, which is the same apparatus that you used i earlier labs. You should review the pre-lab iformatio for those eperimets if you are ufamiliar with this equipmet. 3. Eperimetal ucertaity, the stadard deviatio, ad sigificat figures Scietists ofte speak of eperimetal ucertaity. What is the meaig of this phrase? The idea ca be cofusig, because a lot of effects i a eperimet ca be attributed to ucertaity. To get a hadle o it, let s start with a provisioal defiitio: eperimetal ucertaity is a quatitative statemet about how well a eperimetal result is kow. The importat part of this defiitio is the words quatitative statemet : eperimetal ucertaity is a umber. Ay eperimet is a comple operatio with at least three compoets: the pheomea uder study, the measuremet apparatus, ad the people makig the measuremets. Each compoet ca ad usually does cotribute to the ucertaity i a eperimet. For eample, i our eperimets with acceleratio of carts, the motio of the cart may be a bit differet every time because of frictio from dirt o the track this is ucertaity i the pheomeo itself; the motio sesor may pick up reflectios from differet poits o the cart as it moves this is ucertaity due to the measurig apparatus; fially, there is a variatio amog people i their skill with eperimetal apparatus. You may have oticed that some groups i your lab always seem to get better results tha others. This last poit is ofte uderappreciated. Maipulatig eperimetal apparatus well takes practice; it is like learig a skill i sports or learig to play a istrumet. How does oe determie the ucertaity? The gold stadard i sciece is reproducibility. If differet people use differet apparatus to measure the same pheomeo ad they all get the same result i may measuremets, the the quality of reproducibility is high, ad the ucertaity would be small. But, i the other etreme, if the same people use the same apparatus to measure the same pheomeo, but do ot get the same result over may measuremets, the reproducibility is low, ad the ucertaity would be large. 2/6
Pre-Lab 4: Tesio & Newto s Third Law Thus, the first task i determiig the ucertaity is to establish reproducibility: repeat the measuremets ad record each result. The collectio of results the raw data will likely have some variatio i them. The accepted way to describe this variatio is by usig the mea ad stadard deviatio of the data set. These are defied by the followig equatios. For a give set of umbers, 2,,, the mea is The stadard deviatio σ is + 2 + + = = i. () 2 2 2 ( ) + ( 2 ) + + ( ) σ = = If our set of umbers is a collectio of measuremets we ca use the otatio = ± σ i= ( i 2 ) (2) i= to summarize the data set. This otatio is ofte iterpreted as, has a value of with a ucertaity of σ. But it meas somethig much more specific that we ca state i terms of a procedure, that is, a operatioal defiitio. For a variety of reasos, multiple measuremets of the same quatity i the same maer usually produce distributios that are ormal i the statistical sese. The theory of statistics proves that if a populatio (which is the set of all possible values that could be) follows a ormal distributio, about 2/3 of the populatio lies withi oe stadard deviatio of the mea, ad it is typical for a sample distributio to have the same property. The operatioal defiitio of the otatio is, If is measured a umber of times, approimately 2/3 (more precisely, 68%), of the distributio will be withi σ of the mea value. Statistical theory goes o to show that 95% of a ormal populatio lies withi 2σ of the mea, ad 99% lies withi 3σ. The quatitative measure of reproducibility is give by the relative size of σ ad. If σ is a lot smaller tha, the the reproducibility is high: there is very little variatio amog the set of measuremets. But if σ is about the same size as, the the reproducibility would be low. Note that the size of σ aloe is ot sufficiet to establish reproducibility. For eample, if a legth measuremet gave σ = 2 mm, this would be a eceptioally small ucertaity i, say, the height of Mout Raiier, but a eceptioally large ucertaity i the diameter of a atom. Because oe caot assess the reproducibility of a measuremet without kowig both the value ad the ucertaity, it is commo use the ratio of the two: the fractioal ucertaity (see below). I may cases, calculatig the stadard deviatio of a set of measuremets is all you will eed to do i order to determie the ucertaity. You should at least get comfortable with that process ad lear to do it quickly. However, this is by o meas the ed of the story. 3/6
There is o eact ucertaity. Pre-Lab 4: Tesio & Newto s Third Law Ay ucertaity calculatio is always a estimate. There is o such thig as a eact ucertaity! Why? There are a umber of reasos: Statistics. The mea ad stadard deviatio are calculated from a fiite (usually rather small) umber of measuremets. If you did the eperimet over, you would ot get the eact same set of umbers, ad so you would get a differet mea ad stadard deviatio (although they should be close). I the statistical sese, what you calculate are kow as a sample mea ad stadard deviatio; you would oly get the populatio mea ad stadard deviatio if you had all possible results (ad there may be a ifiite umber of these). Systematic error. Eve if you kew the mea ad stadard deviatio eactly for measuremets usig oe set of apparatus, it is possible that the same eperimet doe with aother setup would give a differet set of results. Effects that skew all of the measuremets oe way or aother give what is called systematic error. If the amout of skew is kow, the eperimeter ca apply a correctio to the data set, but frequetly the systematic error is poorly kow, ad the ucertaity estimate is larger tha the stadard deviatio. Istrumetal resolutio. At the very least, there is a limit to the precisio of the measurig istrumets. If you have a digital voltmeter, for eample, that shows oly three digits, e.g., +3.26 V, you caot say whether the real voltage is 3.262 volts or 3.264 volts. I this case, the istrumetal ucertaity is take to be betwee 0.005 ad 0.0 volt. You should assume that the istrumetal ucertaity of a digital istrumet is equal to least digit o the display, uless told otherwise. A commo mistake of studets is to assume that the last item, the istrumetal resolutio, is the ucertaity i a measuremet. It is at best a lower boud. If the stadard deviatio of a set of measuremets take with a istrumet is otably larger (a factor of 4 or more), tha the istrumet s resolutio, the the istrumetal resolutio does ot sigificatly affect the eperimetal ucertaity. The istrumetal resolutio ca oly be take as the ucertaity i a measuremet if every readig take is eactly the same (or more specifically, the stadard deviatio is smaller tha the smallest digit). Do ot fall ito this trap! (A less commo mistake occurs whe the istrumet really does determie the ucertaity ad every readig is the same i this case, if you were to calculate the stadard deviatio, it would come out to zero! Ucertaity is ever zero.) If the stadard deviatio is about the same as the istrumetal resolutio, the the eperimetal ucertaity should be icreased by about a factor of.4 for that set of measuremets. Because ucertaity calculatios are estimates, they are ever stated with more tha two digits. I other words, σ = 0.3m/s 2 is okay, as is σ = 0.7 m/s 2, but you should ever write σ = 0.32 m/s 2. Fractioal or percet ucertaity. I Lab you estimated agreemet betwee two umbers by calculatig a percet differece. Later you calculated agreemet by lookig at the overlap of two distributios. Let s brig these two ways of lookig at eperimetal results together. We ca defie a fractioal ucertaity ε by the followig formula: σ ε =. (3) Note: σ may represet the stadard deviatio or represets a more geeral ucertaity that would iclude istrumetal resolutio or supposed systematic error. 4/6
Pre-Lab 4: Tesio & Newto s Third Law Whe epressed as a percet (multiply by 00), this is a coveiet way to quickly commuicate the ucertaity i a result, e.g., The measured acceleratio of gravity is 9.8 meters per secod squared, with a ucertaity of 4 percet. Stated i this way, oe ca immediately see whether two results that are compared by their percet agreemet really agree i a quatitatively sesible way. If I kow that my measuremet of g has a percet differece with a epected result of 2% ad my fractioal ucertaity is 4%, the I kow that my data distributio overlaps the epected result: they agree. A importat poit, ofte overlooked by studets, is that fractioal or percet ucertaity ε is uitless, but absolute ucertaity (that is, plai old ucertaity σ ) has the same uits as the value it refers to. This should be obvious from Eq. (3). Sigificat figures. The rules for sigificat digits are as follows :. The leftmost ozero digit is the most sigificat digit. 2. If there is o decimal poit, the rightmost ozero digit is the least sigificat digit. 3. If there is a decimal poit, the rightmost digit is the least sigificat digit, eve if it is a 0. 4. All digits betwee the least ad most sigificat digits are couted as sigificat. The ucertaity of a eperimetal umber is closely related to the umber of sigificat figures used to represet it. Take the value 9.8 m/s 2. That there are three sigificat figures implies a real distictio betwee 9.8 ad 9.84 m/s 2. We assume that the ucertaity i the umber is 0.005 m/s 2. If a eperimet gives a ucertaity larger tha the least digit, the umber of sigificat digits may eed to be reduced. For eample, if the ucertaity were 0.2 m/s 2, the istead of 9.8 m/s 2, g should be stated as 9.8 m/s 2. The geeral rule is that the value ad the ucertaity should be stated with the same precisio, that is, the same umber of decimal places. Here are some eamples of correct ad icorrect ways to write a umber with ucertaity. L is the measuremet of a legth of strig. OK: L = 382 ± 2 mm L = 382.4 ±.2 mm NOT OK: L = 382 ±.2 mm (too may sig. fig. i the ucertaity) L = 382.4 ± mm (too may sig. fig. i the umber) L = 382000. ± 000. µm (too may sig. fig. i uc., ote the decimal poits) To avoid the last bad eample, switch to scietific otatio: 5 L = (3.82 ± 0.0) 0 µm Note that whe scietific otatio is used, the power of te is applied to both the value ad the ucertaity together. Note also that the uits come after all of the umbers. Take from Data Reductio ad Error Aalysis for the Physical Scieces, 3 rd ed., P. R. Bevigto ad D. K. Robiso (McGraw-Hill, New York, 2003), p. 4. 5/6
Pre-Lab 4: Tesio & Newto s Third Law NOT OK: Questio 4 5 5 L = 3.82 0 ± 0.0 0 µm (power-of-te should apply to both umbers together) 5 L = 3.82 0 µm 5 ± 0.0 0 µm (uit label should be at ed of all umbers) A measuremet of acceleratio a of a cart o a iclied track is made with a istrumet that has a resolutio of 0.0 m/s2, ad the data set is used to calculate a mea ad stadard deviatio. The mea is.82 m/s 2, ad the stadard deviatio is 0.07m/s 2. a. What is the operatioal meaig of the statemet a =.82 ± 0.07 m/s 2? b. What is the fractioal stadard deviatio? c. The epected value has a percet differece with the mea of 5%. Does it agree? d. Does the istrumet s resolutio of 0.0 m/s 2, make the ucertaity larger, smaller or the same as the stadard deviatio? Questio 5 The followig measuremets of g were foud i a eperimet. Trial g (m/s 2 ) 9.4 2 9.54 3 0.0 4 9.49 5 9.82 6 9.90 7 9.74 8 9.63 9 9.67 0 9.98 a. Use your calculator or spreadsheet to fid g ad its ucertaity. Write your aswer i the form g = g ± σ g usig correct sigificat digits ad uits. b. Calculate the fractioal stadard deviatio. DBP (0/27/200) 6/6