PAP ER S

www.iop.org/journals/physed

Conceptual and laboratoryexercise to apply Newton’s secondlaw to a system of many forcesCarl E Mungan

Physics Department, US Naval Academy Annapolis, MD 21402-1363, USA

E-mail: mungan@usna.edu

AbstractA pair of objects on an inclined plane are connected together by a string. Theupper object is then connected to a fixed post via a spring. The situation isfirst analysed as a classroom exercise in using free-body diagrams to solveNewton’s second law for a system of objects upon which many differentkinds of force are acting (string tension, spring force, gravity, normal force,and friction). Next, the setup is replicated in the laboratory using rolling cartswith attached force sensors (to measure the string and spring forces) and amotion detector (to measure the position, velocity and acceleration of theobjects). After characterizing the rolling friction, cart masses, incline angleand spring constant, the kinematics and dynamics of the system can beaccurately modelled with no free parameters. Representing the data indifferent ways, notably plotting quantities as a function of the displacementof the carts instead of elapsed time, greatly assists in their interpretation. Forexample, the acceleration of the carts lies along two straight lines whenplotted in that way, the mechanical energy has a zigzag shape and thevelocity of the carts traces out a set of joining half-ellipses in phase space.

Introduction

In lectures in introductory physics, students areintroduced one by one to a variety of forces,including gravity, normal force, friction, tensionin a string and spring force. Simple modelsfor reasoning about and making calculations withthese forces are presented. The importance of free-body diagrams (FBDs) for linking these forcesto the kinematics describing the motion of anobject is emphasized by instructors, and yet moststudents see the construction of such diagramsas an unnecessary extra burden. Often theyinstead tackle problems simply by finding a solvedexample in their textbook and making minor

changes to the resulting formulae to adapt it totheir situation. An FBD does not appear to themto be a useful tool for problem solving becausethey can often guess what to plug into the left-hand side of F = ma when few forces act on anobject. To show the true utility of an FBD, it isnecessary for students to work on exercises wherea rich combination of forces acts simultaneously.Such a situation is presented in this article.

Considering next the laboratory portion ofthe introductory course, students typically performexperiments that individually treat horizontalkinematics, inclined planes, oscillatory motion,Atwood-type setups and at least some aspects

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Conceptual and laboratory exercise to apply Newton’s second law to a system of many forces

of retarding forces (static and sliding friction,and perhaps simple speed-dependent drag forces)using motion detectors and force sensors. Thisvariety of experiments affords the opportunity fora summative laboratory experience in which all ofthese ideas are combined in a single experimentalsetup. To maximize its pedagogical benefits,this capstone laboratory is here integrated withthe conceptual exercise of Newton’s second lawdescribed in the preceding paragraph. It isparticularly satisfying that no free parameters areneeded to model the experimental results. By thisstage of the course, students have built up theirunderstanding of the concepts to the point that theycan successfully account for all observable effects.Physics really does work!

At least two other side benefits of thelaboratory exercise described in this article alsoaccrue to students. First, it is a convincing exampleof the power of alternative representations inunderstanding data. Used in their conventionalmanners, motion detectors and force sensorsrespectively measure kinematic quantities andforces as functions of time. It will turn outhere that it is much more useful to instead plotthem as functions of position. Further insightscan be obtained from phase-space plots and fromenergy diagrams. Secondly, the laboratory workpresents a special opportunity to help studentsto develop a richer understanding of frictionaleffects. In particular, the concept of rollingfriction, important in many real-world devices butoften neglected in the physics curriculum, plays acritical role here.

The article is organized into two majorsections, geared to an Advanced Placement orA-level physics course in a secondary school orintroductory college course. The first section isthe core material, while the additional theory andanalysis presented in the second section are mainlyfor enrichment and further exploration.

The free-body diagrams and experimentTwo objects are investigated that are simultane-ously subject to a variety of forces. The goals ofthe study are to improve students’ conceptual un-derstanding of how to solve Newton’s second lawfor a reasonably complex situation and to enrichtheir physical intuition by then examining a simi-lar setup in the laboratory.

Figure 1. Sketch of the setup for the classroomexercise.

Classroom exercise

Begin with the following problem that groupsof 2–3 students should work on together. It isassumed that the relevant individual forces havealready been discussed previously in the courseand that the students have some experience withconstructing FBDs. In my own classes, I requirethat FBDs include arrows (carefully distinguishedfrom the force arrows) labelled a and v (or thenotations ‘a = 0’ and ‘v = 0’ if appropriate)and x and y to denote the positive directions ofacceleration, velocity and the coordinate axes foreach object of interest. The known directionsfor these four quantities should be used if theyare given or implied in a problem; otherwise,any reasonable choice can be adopted. As willbe apparent below, omitting these arrows onan FBD makes it much harder to successfully‘translate’ the diagram into equations describingthe components of Newton’s second law.

Two blocks are connected by a string and aresliding on a plane inclined at angle θ relative to thehorizontal, as drawn in figure 1. The coefficientof kinetic friction between either block and theplane is μk. The upper block has mass m1 andis connected to a fixed post by a spring of stiffnessconstant k. The lower block has mass m2. Theblocks are released from rest with the spring atits relaxed length (i.e. the length it has beforeattaching the blocks to its lower end). Draw FBDsof each block at some instant in time when thespring has stretched a distance d , assuming thatboth blocks are in motion at that instant. Then use

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your FBDs to find an expression for the tensionin the string connecting the two blocks at thatinstant (expressed in terms of givens and knownconstants, possibly including θ , μk, m1, m2, g, kand d).

The usual idealizations about the forces canbe assumed:

• the string is massless and cannot stretch;• the spring is massless and obeys Hooke’s

law;• the frictional force on a block equals μk

multiplied by the normal force of the planeon the block.

However, rather than reminding studentsabout these concepts in a pronouncement from thefront of the room, it would be better to let groupsdiscuss ideas among themselves. The instructorcan circulate around the room, giving help togroups that want it.

Evidently the blocks are going to oscillateup and down the incline, and eventually come torest somewhere below their starting points. TheFBDs are snapshots of the blocks at some instantof their motion. Students need to realize that thedetails of the FBDs depend on whether the blocksare moving up or down the plane at the instantof interest. So after some time of group work,a class discussion should be held in which thevarious relevant forces should be listed (by name)on the board, together with anything that can besaid about their magnitudes and directions. Theclass may eventually wish to agree on a particulardirection for the velocities of the blocks, for thesake of definiteness. They can then return to theirgroups to continue the exercise.

Here is a polished solution. Choose acoordinate system with x pointing down theincline parallel to its surface and y pointingperpendicularly away from the incline. Assumingthat the blocks are sliding down the incline,figures 2(a) and (b) are FBDs for blocks 2 and1, respectively. In the y directions, the forcecomponents must balance since the blocks canonly accelerate parallel to the surface of theincline. Thus

N1 = m1g cos θ ⇒ f1 ≡ μk N1 = μkm1g cos θ

(1a)and similarly

N2 = m2g cos θ ⇒ f2 ≡ μk N2 = μkm2g cos θ.

(1b)

θ

Figure 2. Both blocks have the same acceleration a and speed v and are acted upon by the same tension T since they are connected by an ideal string. The same coordinate system is chosen for both blocks. The normal forces are indicated by N, the frictional forces by f and the gravitational forces by mg with subscripts 1 and 2 for the two blocks. The spring force on block 1 is Fs. It is assumed that v and a point down the incline for specificity.

TN2

f2

m2g

(a) (b)

m2

θ

Fs

T

N1

f1

x

a

v

y

m1g

m1

Noting that the magnitude of the spring force isgiven by Fs = kd at the instant of interest, the x-component of Newton’s second law can be writtenfor block 1 as

T +m1g sin θ −μkm1g cos θ −kd = m1ax (2a)

and for block 2 as

m2g sin θ − T − μkm2g cos θ = m2ax . (2b)

Note that a must bear the subscript x . Withoutthe subscript, a denotes the magnitude of theacceleration vector a and therefore cannot benegative. With the subscript, ax is the x-component of a and can have either sign [1]. It isagain worth interrupting the groups to have a classdiscussion about the signs of the x-components ofthe velocity and acceleration. Downhill sliding ofthe blocks means that vx > 0 but not necessarilythat ax > 0. The sign of ax depends on whetherthe blocks are speeding up or slowing down as theyprogress down the incline. Without getting into allthe details of the stretch of the spring and the roleof friction, it is intuitively clear that the blocks willbe speeding up when they are high up the incline(e.g. just after release) and slowing down whenthey are far down the incline (i.e. about to turnaround and start climbing back up it).

Based on this class discussion of ax , itshould be apparent that there are some subtleties

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involved with it and hence further study of theacceleration is deferred to the laboratory. Thepresent classroom exercise only asks students tofind an expression for the tension T when thespring is stretched by d . The key insight for thispurpose is that equations (2a) and (2b) are twoequations in two unknowns (ax and T ) and thuscan be solved simultaneously by any of severalmethods, giving groups the opportunity to exercisesome creativity. One way to do so is to divide theleft-hand side of equation (2a) by m1 and the left-hand side of equation (2b) by m2 and equate themto eliminate ax . Solving the result for T then gives

T = m2

m1 + m2kd. (3)

This final expression for T is simple and concrete.It might be worthwhile to check it by showingthat it has the correct units and that it reduces tothe expected result for cases such as m1 = 0,m2 = 0 and m1 = m2. At first sight, it seemsremarkable that equation (3) is independent of θ .Intuitively one might have expected the tensionto increase with increasing tilt of the track. Butkeep two points in mind. First, it is the spring thatultimately drives the motion: equation (3) showsthat the tension T is directly proportional to thespring force kd. Second, equation (3) only appliesif θ is larger than the value θmin = tan−1 μs,because the x-component of the gravitational forcem totg sin θ must exceed the static frictional forceμsm totg cos θ with the blocks (of total mass m tot =m1 + m2) in their initial position when the springis unstretched, or otherwise they will never beginto move! Thus the tension actually does dependon θ in the sense that it is zero for θ � θmin (evenfor an ideal frictionless system, some initial tilt ofthe track is thus required) and it only attains thevalue given by equation (3) after the blocks beginto move and any initial slackness in the string isremoved.

In any case, the real goal of the exercise isnot to obtain equation (3), but to draw the FBDsand properly use them to write down and workwith Newton’s second law. With that in mind, andin preparation for the follow-up experiment, it ismore important that groups can confidently drawfigure 2 and write down equations (2a) and (2b)than that they can simultaneously solve the lattertwo equations.

Laboratory preparation: acceleration as afunction of spring stretch

So far, simple expressions for the spring forceFs = kd and for the string tension T = m2kd/m tot

have been found as a function of the stretch dof the spring from its relaxed position. Neitherof these forces depends on the direction of thevelocity or acceleration of the blocks, and they canbe straightforwardly measured in the laboratoryusing a pair of force sensors and a motiondetector, as described in ‘Laboratory experiment:setup, characterization of parameters and datacollection’. To add richness to the laboratoryexperience, the acceleration of the blocks can alsobe determined by the motion detector. Unlike Fs

or T , when the acceleration is plotted versus d thedata points are found to lie along two straight linesrather than a single one. The theory behind thisresult is described here. Instructors who prefer a‘discovery’ approach in the laboratory may wishto withhold presenting this theory until after themeasurements have been performed.

Anticipating the result, from now on I willsubscript the acceleration a with ‘down’ or ‘up’,depending on whether the blocks are sliding downor up the incline, respectively. Let us start with thecase of downhill sliding, as treated in ‘Classroomexercise’. The x-component of the accelerationcan be found by adding equations (2a) and (2b)together to eliminate T . Rearranging then leads to

adown, x = g(sin θ − μk cos θ) − kd/(m1 + m2).

(4)We see that adown, x is positive only if d is less thansome critical value dc given by setting the right-hand side of equation (4) to zero,

ddown,c = (m1 + m2)g cos θ

k(tan θ − μk). (5)

As a technical aside, which is probably worthmentioning in class only if some student explicitlyasks about it, ddown,c is necessarily positivebecause tan θ > μs � μk. It must be the case thattan θ > μs if the blocks are to begin moving afterthey are released from their initial positions, asmentioned in ‘Classroom exercise’. Furthermore,the static and kinetic coefficients of friction mustphysically satisfy the relation μs � μk. Otherwisea block placed at rest on an incline with angle θ

given by tan−1 μs < θ < tan−1 μk could neitherslide nor stay at rest!

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It is left as an exercise for the reader (or theirstudents) to repeat the preceding analysis for thesituation when the blocks slide up the incline. TheFBDs in figure 2 remain the same except that wemust reverse the three arrows for the directionsof v, f1 and f2. Equations (1a) and (1b) remainunchanged, but in each of equations (2a) and (2b)one must change the sign in front of μk. Replacingax with aup, x , equations (4) and (5) then become

aup, x = g(sin θ + μk cos θ)−kd/(m1 + m2) (6)

and

dup,c = (m1 + m2)g cos θ

k(tan θ + μk). (7)

Comparing equations (4) and (6), one sees thatthe acceleration of the blocks when plottedagainst d is predicted to describe two straightlines with equal (negative) slopes but differentintercepts, one for upward and the other fordownward velocities along the incline. In‘Laboratory experiment: setup, characterization ofparameters and data collection’, this prediction isquantitatively verified in the laboratory.

Laboratory experiment: setup, characterization ofparameters and data collection

An attempt was made to reproduce the setup infigure 1 with standard equipment available in anintroductory physics laboratory. However, evenwhen relatively smooth blocks and incline wereused, kinetic friction was found to rapidly dampout the motion and to occasionally cause theblocks to veer sideways rather than remaining ona strictly linear trajectory up and down the ramp.In order to ensure smooth one-dimensional motionlasting at least a minute (for manageable inclineangles), the blocks were replaced with rolling cartson a grooved aluminum track. This replacementmeans that the sliding friction becomes rollingfriction, and consequently the coefficient of kineticfriction μk will from now on be replaced bythe coefficient of rolling friction μ. A carefultreatment of the theory of rolling friction [2] showsthat one can model the friction by the same formused in equations (1), namely f = μN . Thereforeit is not important that students understand thedetails of rolling friction to conduct and interpretthe present experiment. In fact, at least twoarticles [3, 4] that use the same apparatus as is

used here do not even bother to change the nameor subscript of the frictional force, but continueto call it kinetic friction with coefficient μk. Itwould be preferable to at least alert students to theidea of rolling friction, but from the point of viewof modelling the experimental results it actuallymakes no difference.

The list of equipment for the presentexperiment using rolling carts is given inthe accompanying sidebar, for the benefit ofinstructors who wish to replicate it in theirschools. For specificity, PASCO model numbersare given for some of the key apparatus, but similarequipment from Vernier, PHYWE or other vendorscan also be used.

There are two sets of procedures that studentsneed to follow to obtain data that can be fullycompared to the theory presented in ‘Laboratorypreparation: acceleration as a function of springstretch’ with no free parameters. Most ofthese experimental steps are relatively standardprocedures that students should be familiar withfrom previous laboratory work in the course. Thusinstructors can choose how detailed to make theinstructions they give to the laboratory groups,ranging from a detailed recipe to only a generaldiscussion of the available equipment and goals.

The first set of procedures consists inquantitatively characterizing the experimentalparameters. In brief, we need the values of θ , μ,m1, m2, g, k and the equilibrium stretch s of thespring (beyond its relaxed length) with the cartsattached and at rest. The value of g can eitherbe assumed to be 9.8 m s−2 (which is what I didand which is a reasonable value at the latitudeof my school), or a value can be determinedfrom the standard geodetic formula or from anyindependent experimental method of choice.

Attach force sensors to the top of each cart.Label them ‘1’ and ‘2’ in pencil to keep track ofwhich is which. Weigh them to determine m1 andm2. I found m1 = m2 ≡ m = 0.8319 ± 0.0006 kgsince I used identical carts and force sensors. (Anaccurate electronic scale was used to measure themasses. All other quantities reported in this articlewere obtained from fits or from visually inspectinglengths using a metre stick and consequently haveprecisions of only 1–3%, with correspondinglyfewer reported significant figures.)

Calibrate the stiffness constant k by attachingone end of the spring to a vertical hook and freely

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Conceptual and laboratory exercise to apply Newton’s second law to a system of many forces

Figure 3. Experimental measurements (red dots) to determine the spring constant k from a linear fit (blue line).

Mg

(N)

1.5

0.5

4.0

3.0

2.0

1.0

00 0.2 0.4 0.6

Δy (m)

0.8 1.21

3.5

2.5

hanging known weights Mg from the other end.Use a metre stick to measure the resulting changesin length of the spring, �y. Use a sufficient varietyof weights that the stretch of the spring spans therange of distances that it will undergo on the trackin the actual experiment. I found that hanging from50 to 500 g masses in 50 g increments workedwell. A graph of the resulting values of Mg versus�y is presented in figure 3. A linear fit gives aslope of k = 3.5 ± 0.1 N m−1.

Afterwards, the relaxed spring will probablybe permanently slightly elongated in length, withadjacent coils no longer touching each other.Measure that relaxed length si , which in my casewas 12 cm. We will use this value later todetermine s.

Next the coefficient of friction is measured.Level the track and clip a motion detector to itsend. Choose either of the carts and arrange theattached force sensor’s electric cable on top of it sothat it is out of the way. Starting from the far end ofthe track, give the cart a push towards the motiondetector and collect data on its position. Catch thecart before it hits the motion detector. Plot thespeed squared v2 versus the position x of the cart.(One easy way to do this within LoggerPro is toinsert a manually calculated column into the datatable equal to velocity multiplied by velocity.) Fita straight line to the portion of the data while thecart is smoothly in motion. Divide the resultingslope by 2g to find the value of μ. The theorybehind this procedure is simple. The work–

spee

d sq

uare

d (m

2 s–

1 )

0.10

0.15

0.05

0.001.0

position (m)

linear fit for latest speed squared v2 = mx + b m (slope): 0.1269 m2 s–1 m–1

b (y-intercept): –0.03873 m2 s–2

correlation: 0.9995 RMSE: 0.001500 m2 s–2

1.50.5

kinetic-energy theorem implies that

12 mv2 − 1

2 mv20 = −μmg�x (8)

where �x is the (positive) distance moved by thecart starting from its initial position x0 (when ithas speed v0). Thus �x = x0 − x because thecart is moving towards the motion detector so thatx < x0. Hence a graph of v2 versus x has apositive slope of 2gμ. The result is independentof the mass m of the cart and thus should applyto either cart, assuming that both have the sameconstruction (and neither has been mistreated), butit only takes a few extra minutes to roll the othercart on the track and recollect the data to make surethat both carts give the same slope.

Figure 4 presents typical data for a low initialspeed of a cart, with the analysis performeddirectly in LoggerPro. The slope implies μ =0.0065 ± 0.0002. The same slope was foundfor high initial speeds of the cart (up to at least0.7 m s−1), proving that the coefficient of rollingfriction is independent of speed, as was implicitlyassumed to be the case in deriving equations (4)and (6). This result concludes the characterizationof the experimental parameters.

The second set of procedures is to set up thetrack and carts as depicted in the photograph infigure 5, and to collect kinematic and dynamic datausing them. Notice the considerable stretch of thespring from its relaxed length of si = 12 cm.With the two carts attached and at rest, the loadedlength of the spring was found to be s f = 90 cm.Therefore its equilibrium stretch is s ≡ s f − si =78 ± 1 cm. This number is used to convert values

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C E Mungan

Figure 5. Experimental setup for measuring the forces on and kinematics of two rolling carts.

of position x measured by the motion detector atthe bottom of the track into stretches d of thespring. The angle finder at the top of the track wasused to set the inclination at approximately 10◦above the horizontal. But a more accurate valueof θ = 9.7◦ ± 0.1◦ was computed from the inversetangent of a 37.5 cm rise of the track measuredover a 218.5 cm run using a metre stick.

LoggerPro was set up to collect three graphsas a function of time: force sensor #1 (measuringthe spring force Fs), force sensor #2 (measuringthe string tension T ), and the motion detector(measuring position x , velocity v or acceleration aof the lower end of cart #2). The data collectionwas adjusted to measure 40 samples per secondfor 15 s. To reduce the noise, the number ofpoints for derivative and smoothing calculationswas chosen to be 29. The two force sensors wereindividually calibrated by hanging 500 g from their

hooks vertically. They were zeroed by placingthem on the track in their correct orientations withnothing attached to their hooks.

The spring and string were then attached tothe carts. (The electric cables running from theLabPro interface to the two force sensors werearranged so that they did not snag on any obstaclesduring the oscillations of the carts.) One end ofthe spring ran to a horizontal post at the top ofthe track and the other end ran to the hook onforce sensor #1. One end of the string was tiedto the back of force sensor #1 and the other endran to the hook on force sensor #2. The spring andstring were centred both horizontally and parallelto the track. The motion detector was clipped ontothe bottom end of the track. It was then zeroedwith the carts at rest. It therefore follows that thestretch of the spring is equal to d = s − x , with xoscillating about zero as can be seen in the typical

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Conceptual and laboratory exercise to apply Newton’s second law to a system of many forces

Figure 6. Typical data run in LoggerPro indicating graphs of properly calibrated and zeroed values of x, Fs and T (from top to bottom) over a 15 s interval.

posi

tion

(m)

forc

e 1

(N)

forc

e 2

(N)

0.5

0.0

-0.05

0 5 1510time (s)

time (s)

time (s)

4

3

2

1

0.5

1.0

1.5

2.0

0

0

5 1510

5 1510

data run shown in figure 6. To collect these data,cart #2 was pulled down the track to within about15 cm from the motion detector and released. Thenthe collect button was clicked. Note from the topgraph in figure 6 that data collection started afterthe carts had already climbed most of the wayup the incline, thereby eliminating any distractinginitial features due to the start of motion. Alsothe measurements stop while there is still enoughmotion of the carts that the data have not becomeoverly noisy.

The position graph (top pane in figure 6)has the shape expected for damped sinusoidalmotion. However, instructors should not jump tothe conclusion that the detailed functional form ofthis curve is

x(t) = Ae−γ t cos(ω′t + φ), [WRONG!] (9)

where γ ≡ b/2m tot and ω′ = (k/m tot − γ 2)1/2,appropriate to simple harmonic oscillations of anobject of total mass m tot ≡ m1 + m2 in thepresence of a resistive (drag) force −bv that islinearly proportional to the speed of motion [5].In our case, the resistive (rolling frictional) force

is independent of the speed of motion and hencea different expression for x(t) results [6], as isdiscussed in ‘Laboratory results: basic analysis ofthe kinematics and dynamics’.

The two forces plotted in the lower panels offigure 6 are proportional to d and hence to x . Sincethe acceleration equals the second derivative of theposition x which is sinusoidal, it follows that theforce graphs are phase shifted by 180◦ relative tothe position graph, i.e., troughs of the top graphcoincide with crests of the other two graphs andvice versa. A student can quickly change the topgraph within LoggerPro into a plot of acceleration(rather than position) versus time, and then allthree graphs will have the same phase. The largerlevels of noise seen in the bottom compared tothe middle graph probably originate both from theexpanded vertical scale, noting that equation (3)predicts that T should be equal to half of Fs sincem1 = m2, and from vibrations in the wheels ofcart #1 to which the string is attached.

The data collection can be repeated with slightvariations in the release of the carts, time delayuntil the collect button is clicked, exact pointingof the motion detector and parameters in the

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software until one is satisfied with the results.They should then be saved to a file for the analysisdetailed in ‘Laboratory results: basic analysis ofthe kinematics and dynamics’. Although thatanalysis can be performed within LoggerPro itself,it is easier to use a graphing software package suchas a spreadsheet.

Laboratory results: basic analysis of thekinematics and dynamics

So far, the data for the kinematics and forceshave been plotted in LoggerPro as a function oftime. But according to the theory presented in theprevious subsections, it is easier to model the dataif we replot them versus position instead of time.Since LoggerPro saves six columns of data bydefault (corresponding to t , F1 = Fs, F2 = T , x ,vx and ax ) it is easy to do this within a spreadsheetsuch as Excel by selecting the two desired columnsof data. The results are plotted in figures 7(a)–(c)as the blue dots. The theoretical predictions aregraphed by the solid lines, using the values of theparameters determined in ‘Laboratory experiment:setup, characterization of parameters and datacollection’ (allowing them to vary at most withintheir specified precisions).

Figure 7(a) plots the spring force. Recallingthat d = s − x , the model predicts that

Fs = k(s − x), (10)

shown as the red line, in excellent agreement withthe data. Panel (b) graphs the tension in the stringconnecting the two carts together. Equation (3) canbe rewritten in terms of x as

T = m2k

m1 + m2(s − x), (11)

which is shown as the red line and again describesthe measurements well. Finally, figure 7(c) plotsthe acceleration of the carts as a function of theirposition. In striking contrast to the plots in panels(a) and (b), the acceleration data lie along twostraight lines, rather than only one. These two linescorrespond to motion of the carts up the incline(corresponding to the red curve) and down theincline (green curve), with turn-arounds occurringat extremal values of a (when v = 0). Thesetwo theory curves use the mean values of all ofthe measured parameters with no undeterminedcoefficients. Since the motion detector is located

Figure 7. (a) Graph of the measured spring force Fs versus the position x of cart #2 (blue dots), together with a plot of the theoretical prediction (red line). (b) Graph of the measured tension T in the string versus the position x of cart #2 (blue dots), together with a plot of the theoretical prediction (red line). (c) Graph of the measured acceleration ax of the carts versus the position x of cart #2 (blue dots), together with a plot of the theoretical prediction (red and green lines).

spri

ng f

orce

(N

)te

nsio

n (N

)ac

cele

ratio

n (m

s–2

)

1.0

2.0

3.0

4.0

5.0

–0.6 –0.4 –0.2 0 0.2 0.4 0.6position (m)(a)

(b)

(c)

0–0.6

–0.4–0.9

–0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4

–0.6

–0.3

0

0.3

0.6

0.9

1.0

2.0

0.5

1.5

2.5

–0.4 –0.2 0 0.2 0.4 0.6position (m)

position (m)

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at the bottom of the incline, it measures ax > 0for accelerations directed up the incline, oppositeto the convention of figure 2, and so we reverse thesigns of equations (4) and (6) to obtain

adown, x = −g(sin θ − μ cos θ)

+ k(s − x)/(m1 + m2) (12)

for the green curve, and

aup, x = −g(sin θ + μ cos θ)

+ k(s − x)/(m1 + m2) (13)

for the red curve. The difference in the interceptsof these two curves is

adown, x − aup, x = 2gμ cos θ (14)

and thus the vertical separation between the twolines is a sensitive measure of rolling friction. Theexcellent fits demonstrate that μ is independent ofθ and v, and that air drag is negligible (as furtherdiscussed in the appendix), at least for the range ofmotions investigated here.

Further analysis of the experimental dataSo far the spring force Fs, the string tension Tand the acceleration of the blocks ax have beenanalysed as a function of position. One could stophere. However, there are additional interestingaspects of the data that can be mined from them, iftime and student interest permit. Specifically, threeother representations of the data will be discussedin this section.

Time dependence of the position

I have already argued that equation (9) is wrongbecause it only applies when the drag is linear inthe speed of the object. In the present experiment,the frictional force is independent of the speed ofthe carts. It is not hard to conceptually deduce thecorrect expression for the resulting position of thecarts as a function of time [7], although it requiresa bit of care to write it down mathematically [8].

Consider some half-cycle of the motion ofthe carts, during which they are rolling down theincline. Throughout that interval of time (betweensuccessive zero-crossings of the velocity), thesystem of two carts (treated as a unit) is subjectto only three force components along the incline:the spring force Fs, the x-component of the totalweight m totg of the two carts and the sum ftot =f1 + f2 of the frictional forces on the carts. If

the only force were Fs, then the carts wouldoscillate with constant amplitude A about therelaxed position d = 0 of the spring. Next imagineturning on gravity without friction. This situationcorresponds to the familiar classroom discussionof a mass on a horizontal spring that is rotateduntil it is hanging vertically. The amplitude Aof oscillation is unchanged; the only effect ofturning on gravity is to shift the centre point ofthe oscillations to the new equilibrium positiond = s. Finally, turn on the rolling friction. Sinceit is a constant force, its effect on the motionis analogous to gravity’s effect (which is also aconstant force). Namely the centre point of theoscillations is again shifted.

We can now mentally trace out the motion ofthe system. Start with the carts momentarily atrest near the top of the incline at d = 0 for thefirst half-cycle of the motion; that position is a‘top turning point.’ The carts will begin to rolldownhill. Eventually they will pass through thecentre point of their downward motion, at whichposition the sum of the three force componentsalong the incline is zero. At that centre point, d =ddown,c as given by equation (5), with the kineticcoefficient μk replaced with the rolling coefficientof friction μ. The difference in position between aturning point and the equilibrium point defines theamplitude of motion. Consequently the amplitudeof the carts’ motion for the first half-cycle is A1 =ddown,c. The carts will continue to roll downhilluntil they reach a ‘bottom turning point’ at a totalspring stretch of double the amplitude, d = 2A1 =2ddown,c.

Next the carts will start rolling back uphill.But when that happens, the frictional forcereverses direction. Consequently the springposition at which the three force components sumto zero changes to d = dup,c given by equation (7).The amplitude for this second half-cycle is thedifference in position between the bottom turningpoint and this new equilibrium point, so that A2 =2ddown,c − dup,c. The amplitude is thus decreasedby �A ≡ A1 − A2 = 2m totgμ cos θ/k. Thistrend continues: the amplitude drops linearly (notexponentially) by �A every half-cycle until thecarts come to rest at one of the turning pointsafter a finite number of oscillations [9]. Infurther contrast to the viscous (speed-dependent)damping of equation (9), the angular frequency ofoscillation is always equal to the undamped valueof ω = √

k/m tot.

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C E Mungan

Figure 8. Graph of the measured position x of cart #2 (blue dots), together with a plot of the theoretical prediction (red curve).

posi

tion

(m)

0–0.6

–0.4

–0.2

0

0.2

0.4

0.6

2 4 6 8 10 12time (s)

Putting these ideas together, the amplitude inthe nth half-cycle has the constant value

An = A1 − (n − 1)�A = m totg

k× [sin θ + μ(1 − 2n) cos θ ]. (15)

Assuming one starts the measurement of time twith the spring at its relaxed length (i.e., d = 0at t = 0), then n is calculated by dividing t byT/2 = π/ω and rounding upward to the nextlargest integer. The computed position of the cartsas a function of time is given by

x(t) = An cos(ωt). (16)

To compare this prediction to the experimentaldata graphed in the top panel of figure 6, we needto shift the origin of time t horizontally to accountfor the fact that t = 0 actually indicates when thecollect button was clicked. The resulting curveis plotted in red in figure 8 and is in excellentagreement with the data points in blue.

Phase-space motion of the carts

Now that we have an accurate model forthe position of the carts, we can differentiateequation (16) with respect to time to get thevelocity of the carts,

vx(t) = −Anω sin(ωt), (17)

where again t is actually calculated as t − t0with t0 being the time the carts would be atrest at the unstretched position of the spring (by

Figure 9. Phase-space plot of the velocity of the carts versus the position x of cart #2 (blue dots), together with the theoretical prediction (red curve).

velo

city

(m

s –

1 )

0

–0.3

0.3

0.6

0.9

–0.6

–0.90 0.2 0.4 0.6–0.2–0.4–0.6

position (m)

extrapolating backward in time). In practice,t0 is found by shifting the calculated curve infigure 8 horizontally until it best coincides with theexperimental data. To be accurate, the carts shouldbe released from rest with the spring relaxed,to ensure that the amplitude of motion matchesequation (15), rather than being released nearthe bottom of the incline as was done in theexperimental procedures described in ‘Laboratoryexperiment: setup, characterization of parametersand data collection’.

A graph of velocity versus position mapsout what is known as a phase-space orbit. Anygiven point on such a curve specifies the initialconditions needed to uniquely determine themotion of the object thereafter. (In general,a solution of Newton’s second law requiresspecification of x0 and v0.) In the presence ofdamping (without a compensating driving force),the orbit will gradually decay towards a pointof zero velocity at some specific final position,without ever crossing itself. Consequently, forreal undriven oscillators the phase-space trajectoryis a clockwise inward spiral [10, 11]. Ourexperimental and predicted curves are plotted infigure 9 and show that behaviour, starting from apoint of large positive position and small positivevelocity near the right-hand edge of the figure.

If one has gone to the work of generatingthis visually appealing phase-space graph in class,it may make sense to follow up by discussingwith students the actual mathematical shape of

284 P H Y S I C S E D U C A T I O N May 2012

Conceptual and laboratory exercise to apply Newton’s second law to a system of many forces

this orbit. During any half-cycle (betweensuccessive crossings of the horizontal central axisin figure 9 when v = 0) the amplitude ofoscillation is a constant, just as it is for anundamped simple harmonic oscillator. But for anundamped oscillator, the phase-space trajectoriesare ellipses [11]. Consequently, each half-cycleof the graph in figure 9 is half of an ellipse,(x/An)

2 + (vx/Anω)2 = 1 from equations (16)and (17). At the nth turning point, the semi-major and semi-minor axis lengths of a givenellipse suddenly decrease by a factor of �A/An

according to equations (16) and (17), but the centreof the ellipse also shifts horizontally so that thereis no discontinuity in the orbit.

Mechanical energy of the system

It is often instructive in studying oscillators toconsider the kinds of energy involved [12]. Inthe present situation, there is kinetic energy of thetwo carts, elastic potential energy of the spring,gravitational potential energy as the carts movealong the incline and thermal energy generated byrolling friction (as the wheels flex against the trackand the bearings rub against the axles). The elasticpotential energy is equal to

Us = 12 kd2 = 1

2 k(s − x)2 (18)

and the gravitational potential energy is

Ug = m totgh (19)

where h is the height of the centre of mass ofthe two carts above some reference level. Thesimplest choice is to measure up from the table tothe bottom of cart #2, since that is what the motionsensor monitors, so that h = x sin θ . Substitutethat expression into equation (19) and then replacem totg sin θ with ks, which must be equal to itbecause forces along the incline balance at theequilibrium position of the loaded spring. Nowadd together the two forms of potential energy toget a total of

U = 12 kx2 (20)

after discarding an unimportant constant of 12 ks2

(which can be eliminated by shifting the referencelevel appropriately). Equation (20) simply statesthe familiar fact that the total potential energy ofthe system is proportional to the square of thedisplacement x from the equilibrium (rather than

Figure 10. Graph of the mechanical energy of the system versus the position x of cart #2 (blue dots), together with the theoretical prediction (red curve).

mec

hani

cal e

nerg

y (J

)

0.6

0.5

0.4

0.3

0.2

0.1

00 0.2 0.4 0.6–0.2–0.4–0.6

position (m)

the relaxed) position of the spring. The totalkinetic energy of the system is

K = 12 m totv

2 (21)

and adding equations (20) and (21) gives thetotal mechanical energy E = K + U .Experimental values of E can be computed fromthe measurements of x and v by the motion sensorand our characterization of the masses and springconstant. The result is plotted versus position x asthe blue dots in figure 10 and has a zigzag shape.

Theoretically it is easy to see how this shapearises. The analogue of equation (8) for the cartsrolling on the incline is

�E = − ftot�x = −μm totg cos θ�x (22)

where ftot is the sum of the rolling frictionalforces on the two carts. Thus we predict that theenergy drops linearly with a slope of magnitudeμm totg cos θ while x reverses direction each half-period of oscillation, as plotted in red in figure 10.Note that if we instead plotted E versus timet , the slope would not be constant and so thepattern would not simply be a straight line foldedback on itself. Equation (22) compactly representsthe conversion of mechanical energy into thermalenergy by rolling friction.

ConclusionsLooking back over our journey in this article, westarted by drawing free-body diagrams in figure 2that are rich enough to challenge students whoare used to skipping that step in solving Newton’s

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second law. From figure 2, we found an expressionfor the tension that is simple and that turns outto be the same for both upward and downwardmotion of the blocks. It only depends on themass ratio m1/m2 and on the spring force kd ,and is independent of the individual masses ofthe blocks, the gravitational field, the angle of theincline and the coefficient of friction. On secondthought, how can that be? In physics instruction,we often emphasize that solutions can be tested byconsidering limiting cases. But clearly the tensionis not given by equation (3) in the limits that theangle of the incline becomes zero, the coefficientof friction becomes very large or in zero gravity.However, while T itself does not depend on θ , μk

or g, its minimum value does for equation (3) tohold, in that we assume the blocks will begin toslide down the incline once released. This exampleserves as a caution against blindly taking limitingcases of a problem solution.

In contrast to the tension, we found that theacceleration of the blocks is different for uphilland downhill motions. To test this prediction,we went into the laboratory and employed cartsinstead of blocks, replacing the concept of slidingfriction with rolling friction. Although the detailedmechanisms of these two kinds of friction aredifferent, both can be modelled as a coefficientμ multiplied by the normal force N on theobject. In particular, the rolling coefficient couldbe measured by pushing a cart along a level track,where the resulting slowing by friction is readilyvisible to students.

Although we were able to analyse theposition of the carts (and hence their velocity andacceleration by differentiating) as a function oftime in the form of equations (15) and (16), itrequired some care. Furthermore the graph infigure 8 does not immediately show the differencebetween motion up and down the track. For thispurpose, it turned out to be much more revealingto plot the acceleration against position insteadof time. Two straight lines then resulted, withthe vertical spacing between them being directlyproportional to the coefficient of rolling frictionaccording to equation (14).

This technique of plotting quantities versusposition also showed itself to be a useful way tointerpret the variation in the spring force, stringtension and mechanical energy of the systembecause each of them is then a straight line (folded

over itself in the case of the energy). Anothervisually enlightening way to graph the motion ofthe system is in phase space. Each half-cycle ofthe motion in figure 9 turns out to describe halfof an ellipse, whose major and minor axis lengthsdecrease by the same linear increment after eachturning point.

The experiments and concepts presented herecan be extended in other directions. Shaw [13]has shown that if a hanging weight drives therolling of a solid cylinder, the frictional force canbe in either the opposite or the same direction asthe translational motion, depending on whetherthe string is wound around a disc whose radiusis a small or a large fraction of the radius ofthe cylinder, respectively. Note that the rollingfriction is negligible compared to the driving forcein Shaw’s experiment, and hence the frictionhe is measuring is static (in the absence ofslipping) not rolling. Finally, getting closer tothe extramural interests of many students, the freerolling (coasting) of vehicles (such as bicycles [14]or cars [15]) can be measured. Air drag becomessizeable once the speed gets high enough (above5–10 m s−1), but at low speeds rolling frictiondominates. The overall resistance depends on tyreinflation, transmission losses and streamlining ofthe shape [16].

Acknowledgments

I thank Midshipman Alexander Pybus in myGeneral Physics I course, whose in-class analysisof figure 1 motivated this article. Thanks alsogo to Ensign James Kelly for discussions andmeasurements of the rolling friction, and toProfessor Michael Faleski from Delta College whosuggested plotting the mechanical energy.

Sidebar: Laboratory equipment list (perstudent station)

• two PASCO collision carts (ME-9454);• two PASCO dual-range force sensors (used

on the ±10 N scale);• one PASCO 2.2 m track (ME-9779);• one PASCO motion sensor (used on the

narrow-beam setting);• one PASCO harmonic spring (ME-9803);• computer with LabPro interface and

LoggerPro software;• posts (to support the track and spring);

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• 25 cm thread (to connect the two cartstogether);

• hanging weights (to calibrate the springconstant and force sensors);

• electronic scale (to weigh the carts with theirattached force sensors, m1 and m2);

• angle finder (to adjust the track to θ ≈ 10◦);• metre stick (to measure the spring stretch and

exact track angle).

Appendix. Air drag on the rolling cartsHere it is shown that air drag can be ignored forthe speeds of this experiment. The rms speed ofthe carts during the 15 s interval in figure 6 iscalculated from the data to be v̄ = 0.40 m s−1. Thecross section of a cart with its attached force sensoris roughly square with an area of approximatelyA ≈ 50 cm2. Consequently the Reynolds numberis large enough that the air resistance is quadratic(rather than linear) in the speed, FD ≈ 1

2 Aρv̄2,where ρ = 1.2 kg m−3 is the density of air [17].We can compare this value to the coefficient offriction measured in figure 4 by dividing by theweight of the cart to get FD/mg = 6 × 10−5. Thisvalue is only about 1% of the value of μ foundexperimentally. Therefore the air drag on the cartsis on average 100 times weaker than the rollingfriction and is consequently negligible.

Received 2 September 2011, in final form 23 September 2011doi:10.1088/0031-9120/47/3/274

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Carl Mungan is an Associate Professorof physics at the US Naval Academy,preparing 4000 officers for service in theAmerican Navy and Marines. Allstudents are required to take at least twosemesters of physics as part of theirpreparation.

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