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Is it simple to explain simple experiments? (The metre-stick experiment)

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2012 Phys. Educ. 47 701

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P A P E R S

www.iop.org/journals/physed


Dragia Ivanov and Stefan Nikolov

Faculty of Physics, Plovdiv University P Hilendarski, 24 Tsar Asen street, 4000 Plovdiv, Bulgaria

E-mail: [email protected] and [email protected]

Abstract

The demonstration of finding the centre of gravity of a metre-stick (or another long thin body) is fairly well known and is performed often due to its simplicity. Explaining the demonstration is not so simple, however, if one wants to consider some of the finer details. In this paper we give an in-depth consideration of the demonstration and the friction forces involved. The process is presented both analytically and graphically. We have also considered the role of the demonstrator as an important factor.

Introducing the problem

Let us perform the following simple and fun experiment. We need a smooth homogeneous stick with a constant cross-section and a sufficient length (at least about 1.5 m for a good demonstration). It can be made of any material wood, plastic, glass, metal, etc. A metre-stick is frequently used [ 1 4]. We place the stick on the outstretched index fingers of both hands non-symmetrically relative to them trying to keep it horizontal (figure 1). We start to slowly move our fingers towards each other. Will we make the stick fall down this way? No matter how much we try, we cannot do this. Many trials show that irrespective of the initial position, the two fingers always come together in the middle of the stick and it stays on top of them in balance. This result seems unexpected.

We found that this well-known experiment has not had a satisfactory in-depth explanation. Previous considerations by others [ 5, 6] have focused on the exact moment and number of

switches with idealized models. We have also treated this experiment in the past [ 7], but did not at the time explain some important details (the lack of simultaneous sliding at the two fingers, the exact mechanism for switching of the fingers). A case for the need of more in-depth realistic consideration can be found in [ 8]. We will try to give an explanation as exhaustively as possible.

0031-9120/12/060701C08$33.00 c 2012 IOP Publishing LtdP H Y S I C S E D U C A T I O N 47 (6)701

D Ivanov and S Nikolov

First let us perform the experiment a little more precisely (figure 2). We determine the middle point of the stick and mark it. Thus we denote the centre of gravity (CG) of the stick (point O). Symmetrically on both sides of CG we make two marks on the stick at a distance of about 50 cm from O. On a horizontal table we place two identical smooth pieces of L-shaped steel (as shown in the figure) that will be the supports A and B. On these supports we place the stick so that the left support A is exactly under the mark and the right support B is further away from the CG than A, i.e. it is to be to the right of the mark. Thus the two supports are asymmetrical relative to the middle of the stick.

After that we take the two supports in our hands and start sliding them towards each other simultaneously, slowly, evenly, without jerks. We observe carefully the movement of the stick and the supports and their relative positioning. At first the stick is immobile relative to the support A (the one closer to the CG) and moves along with it towards the support B (the one further from the CG). The support B reaches the symmetrical mark, passes some distance after it and only then does the stick stand immobile relative to it, at the same moment starting to slide relative to A. Continuing in this way, the events are repeated but the two supports switch roles. After several alternative switches of movement relative to the two supports they meet at the centre of the stick and the experiment is over.

We could perform the experiment in a slightly different way. We can hold one of the supports immobile relative to the table and move only the other one. The stick performs the same movements relative to the supports. The end result of the experiment is the same.We must note that during the trials we never observe simultaneous movement of both supports relative to the stick (as long as we move the supports slowly). This has not been explained before.

On friction forces

In order to explain the results of the experiment we need to use some basic knowledge about friction forces that we will give here briefly.

Let us place a block of mass m on a smooth horizontal immovable surface (figure 3). TheEforce of gravity G and the normal reaction forceER act upon the block. The normal pressure forceEN acts upon the support. (Note: since the twoEEforces R and N are an actionreaction pair as per Newtons third law they are frequently referred to as a single normal contact force. This is okay among experts or students who understand the finer details, but in front of not-so-well prepared students it might be useful to use this occasion to once again reiterate the more subtle nuances of Newtons third lawthe fact that there are two distinct forces involved, acting on two different bodies, frequently with different results.)EE EWe can write down G D R D N, G D R D N. Let us pull the block by means of a horizontal dynamometer, slowly increasing the magnitudeEof the pulling force F from zero. At first, even though the dynamometer indicates a force to be present, the block remains immobile. It means that upon it in a horizontal direction another forceEis acting in an opposite direction to F and the two forces are equal in magnitude at any given moment. This force is called the static frictionEforce FS. It is caused by the interaction between the touching surfaces of the block and the support and depends on the smoothness of the surfaces and the interactions between their molecules.

By increasing the magnitude of the force F, the magnitude of FS increases too. While the block is motionless, we have FS D F. Clearly, the static friction force has a maximal value FSM. When the pulling force F reaches this maximum value, i.e. when F D FSM, the block breaks off

702P H Y S I C S E D U C A T I O N2012


from the surface with a small leap forward, after which it starts to slide. Depending on the nature of the contacting surfaces and the mass of the block, the leap can be significant. The block makes the leap because the pulling force at that moment turns out to be larger than the force of kinetic friction FK. Upon closer observation it can be seen that at the moment the block starts moving the indication of the dynamometer decreases with a small jump and afterwards remains constant and equal to the kinetic friction force FK. Obviously the spatial leap and the leap of force are related. It is important to pay attention to this leap-like transition from a state of rest to a state of sliding during which we have a difference between the magnitudes of the two friction forces, i.e. FSM > FK. As we will see later, this difference, small as it may be, is very important to explain the stick experiment. The experimenter interferes in the sliding block experiment by unconsciously decreasing the pulling force at the moment of the leap from FSM down to FK. Because of this after the leap the block moves at a small constant velocity.

The conclusion from the conducted experi-ments is that the static friction force can vary from zero to a certain maximum value FSM, i.e.

0 FS FSM:

In practice it is sometimes assumed within a certain degree of accuracy that at small velocities the kinetic friction force is equal to the maximum static friction force FSM D FK. In our experiment, however, the difference between FK and FSM is crucial.

In figure 4 the friction forces are represented graphically relative to the pulling force.

It is experimentally determined that FSM is proportional to the normal contact force R and depends on the contacting surfaces (expressed as a static friction coefficient S):

FSM D SR D Smg:

The kinetic friction force is analogously given as

FK D KR D Kmg;

where K is the coefficient of kinetic friction. The static friction coefficient S is always larger than the kinetic friction coefficient K, i.e. S > K.

Balance conditions for the stick

Let us denote the total length of the stick as 2L. In the beginning the support A is at a distance a from the CG and the support B is at a distance b, with 0 a L and 0 b L. Acting upon theEstick are the gravity force G (applied in the CG) and the normal reaction forces at the two supportsEERA and RB, pointing vertically up (figure 5).EWhen we apply external horizontal forces FA andEFB against each other rising from zero to the

EA EB supports, static friction forces FS and FS initiallybegin to act upon the stick, acting horizontally and against each other. These forces under certain conditions can change into kinetic friction forcesEA EBFK and FK.

Here we must discuss a very important pecu-liarity of the system, caused by the participation of the experimenter. Through visual control and the sensory perceptions of the fingers of both hands and with the decisive participation of the brain, the experimenter stresses his muscles in such a way so as to maintain a constant equality ofEEthe two external forces, i.e. jFAj D jFBj. Because of that, in the beginning, when applying small enough forces the two supports and the stick will not move.

Let us now determine the maximum static friction forces and the kinetic friction forces acting upon the stick. We will need them for the subsequent arguments.



In order for the stick to be in balance the algebraic sum of the torques from the forces acting on it relative to an arbitrary axis has to bePzero, i.e. Mi D 0.

Let us now describe the balance condition relative to axes passing horizontally through the points A and B perpendicularly to the stick. (The points A and B are situated exactly above the supports A and B.) Considering the notations in figure 5 and that forces acting towards the axis (e.g. the friction forces) produce zero torque we obtain:

A ! RB .a C b/ G a D 0;

B ! RA .a C b/ G b D 0:Therefore:

RA D G

b

;RB D G

a

:

aCb

aCb

If we assume that the friction coefficient S is the same for both supports, for the magnitudes of the maximum static friction forces we obtain:

FSMAD SRA D SG

b

;(1)

a C b

FSMBD SRB D SG

a

:(2)

aCb

The respective kinetic friction forces are:

FKAD KRA D KG

b

;(3)

a C b

FKBD KRB D KG

a

:(4)

aCb

Let us assume that initially support A was closer to the CG than support B, i.e. a < b. Thus

we have:

FSMA > FSMB:

The results up until now have been essentially the same as those we have published before [ 7]. Considering these results, let us continue the explanation of the experiment.

Explaining the results of the experiment

So from the initial condition when the system is at rest we start to act upon the two supports with gradually increasing forces from zero. As a result, the static friction forces acting upon the stick gradually increase from zero towards their maximum values.

Since FSMA > FSMB, the static friction force at the support B will be first to reach its maximum value FSMB and transforms with a small leap down into a kinetic friction force FKB. At this moment, with the participation of the experimenter the static friction force on the left (which has not yet reached its maximum value) decreases and becomes equal to the kinetic friction force on the right. From that moment the two supports begin to move towards each other. Then the left one stays immobile relative to the stick and carries it along and the right support slides relative to the stick towards the CG. As can be seen from formula ( 4) with the decrease of b, FKB gradually increases. But, as can be seen from formula ( 1), FSMA gradually decreases. The two forces change approximately linearly with the decrease of b, which is not significant in this case.



When support B comes to the symmetrical point on the right of the CG at a distance of b D a the kinetic friction force at B is still smaller than the maximum static friction force at A FSMA. This is because, other conditions being equal, the kinetic friction force is smaller than the maximum static friction force, i.e. FSMA > FKB when a D b. Consequently, at the symmetric position the movement of the support B continues towards the CG and the symmetric position is passed over. How long will this continue? Only at a subsequent position of B, even closer to the CG, do the two forces become equal in magnitude, i.e. FSMA D FKB when c < a. Upon further negligible movement of B towards the CG it turns out that FSMA < FKB. At that point the stick can no longer stay immobile relative to support A and starts sliding to the left relative to it, i.e. support A starts to move towards the CG. But what happens at this moment in respect of support B, relative to which the stick has been sliding up until this moment? Is it possible for the stick afterwards to slide relative to both supports simultaneously?

As we stated in the beginning, the simultane-ous sliding of the stick relative to both supports is never observed experimentally. Why does the stick stop sliding relative to B at the moment when it starts sliding relative to A? This is actually the hardest element of this experiment to explain. We offer two possible explanations.

While we were moving the support B from the distance b to the distance c relative to the CG it was in principal possible to maintain the friction forces as equal by way of our hands, i.e. FSA D FKB. At the moment when the sliding relative to A starts the maximum static friction force FSMA automatically leaps down to FKA, i.e. FSMA > FKB. It is clear that the distances a and c are now significantly different, i.e. a > c. But from ( 3) and ( 4) it can be seen that in this case FKA < FKB. Their difference FKB FKA D

KG aCac KG aCcc D KG aaCcc is directed left towards the support A. This net force acts on the stick during the leap automatically (without participation of the experimenter). A short-time push arises on the stick that moves it to the left with acceleration. This small displacement of the stick is practically indiscernible to the eyes and fingers. Since the support B is moving left at this moment (slowly, as determined) and during the automatic jerk the stick also moves to the left (even if only just slightly), there comes a moment when their velocities become equal. This means that for a moment the stick becomes stationary relative to B.This short-time stop is enough to transform the kinetic friction force FKB into a stationary friction force FSB. The movement of the stick relative to B stops. At this moment the

support A is farther from the CG. Because of that FKA < FSMB.

The described transformations of the forces happen in a short time, at which the movement of the stick changes its nature symmetrically relative to A and B,

i.e. sliding relative to A and stationary relative to B. By means of the hands, the two forces are again maintained equal in magnitude, i.e. when a > b, FKA D FSB.

The change of the motion relative to the two supports can also be caused by the unconscious actions of the experimenter himself. Since the supports have to be moved slowly, the experimenter might occasionally stop them for a moment, without noticing it. Thus he could restart the experiment and change the nature of the movement without the automatic jerk, if the conditions are appropriate (support B has to be closer to the CG than A). While this may be the predominant effect in some real-life experiments the automatic mechanism described above applies even to ideal cases such as those considered in [ 5, 6].

After the change, an analogous movement of the stick is carried on, symmetrical to the one just described and the movement of the stick alternately changes relative to A and B several times. If the supports are very thin, one of them reaches the CG first (for example A). According to ( 4) we can see that when a D 0 we obtain FKB D 0. The friction force at the support B drops to zero and the stick stays immobile on top of support A irrespective of the horizontal position of support B. Since the two supports in fact have finite sizes, when they touch each other the movement of the stick stops and the CG is positioned between them.



There is one theoretical possibility for the simultaneous sliding of both supports under the stick. This could happen if they are initially placed exactly symmetrically relative to the CG and their friction coefficients (both static and kinetic) are exactly the same. This, however, is practically impossible.

For improved clarity we will represent the change of the friction forces acting on the stick graphically (figure 6). In order to do that we will work out some exact quantitative results concerning the motion of the stick and we will use them to draw the graph.

For better clarity when writing down the formulae we will denote the initial starting positions of the supports as a D a0 and

b D b0. In the beginning the two maximum static friction forces are respectively A

b0B

a0

FSMD SGa0Cb0, FSMD SGa0Cb0. The

A

ratio of the two forces is

FSM

b0. If

FSMBDa0

a0 < b0 at the start the stick will start moving relative to B. Since while a0 D const and b0 decreases down to b1 in the first

switching point, according to ( 1) the maximum static friction force alsodecreases. When bDb1its value is

A

b1

FSMD SGa0Cb1. Therefore in the moment

of the switch the ratio of its starting

A

b0

magnitude FSM D SGa0Cb0to its final

A.b0/

magnitude isFSM

b0.a0Cb1/ .

A

FSM.b1/D b1.a0Cb0/

The switching from motion to rest or vice versa in a support happens when the maximum static friction force at one support (the one that had been at rest) becomes equal to the kinetic friction force at the other.

Therefore, when switching support B from motion to rest we have FSMA D FKB, SG aCbb D KG aCab , and thus Sb D Ka,a D S . This means that until the momentbKof the switch a has been constant and b wasdecreasing down to a value of b D K a.SAnalogously, when switching the support Afrom motion to rest FSMBD FKA, SGaD

a b

b

b

SC

KGaCb, hence Sa D Kb,aDK,

K

a D S b.


b D


Let us consider the switch when before it the support A was immobile and B was moving. We showed in the above paragraph that at the moment of the switch in this case K a. Substituting this in ( 1) we obtainSA

b

S KFSM D SGaCb

DSC KG.In the same way we can obtain the valueBS K

of FSM DSC KG for the maximum frictionforce at B for the case of B being immobile prior to the switch and A moving.

This means that all the switches happen at the same value for the friction forces, i.e. on a straight line parallel to the abscissa at a distance of:

y D S K G: S C K

At the instants of the switches all the leaps 1F of the maximum static friction force FSMA down to the kinetic friction force FKA are

FSMA

SA

S K

equal. SinceFKA

DKand FSM D

SC KG,

we obtain: 1F D FSMA FKA D FSMA

K FADK. S K/ G. The same applies to

S SM

SC K

the support B. This means that for all

switches the change of force is the same:

1FD

K. S K/G:

S C K

Assuming that a0 < b0 and support B is the first to start moving, the points of switching get closer according to the relations an D . K /2na0, n D 0; 1; 2; : : : ; bn DS

K /2n 1a0, n D 1; 2; 3; : : :, where an is the

S

distance of A to the CG at its nth switch; bn is the distance of B to the CG at its nth switch. This result follows from:b1 D .K/1a0, a1 D .K/2a0, b2D .K/3a0,

S

4

S

S

a2 D .K/

a0, b3 D .K/5a0, a3D .K/6

S

S

S

a0; : : :. A similar, less rigorous (numerical only) derivation of this can be found in [ 5].

The detailed tracking of the force changes in the switching points when the supports are immobile is quite complicated and has to be done very carefully. We also need to take into account that the hands constantly keep the forces at the two supports equal in magnitude. On figure 7 we have represented in more detail the first two switches on the left and right using the numerical values from figure 6.

The maximum static friction force at A is FSMA.b0/ D 0; 83 N, and at B it is FSMB.a0/ D 0; 67 N (point L1). Since FSMB.a0/ < FSMA.b0/ the sliding first starts at B. Because of that FSA rises

only to 0, 67 N (point K1). When the sliding starts at B the force FSMB drops down from 0, 67 N down to the kinetic friction force FKB.a0/ D 0; 55 N (point L2). This leads to a corresponding decrease of FSA down to 0, 55 N (point K2). Support B slides from b0 to b1, at which the force FKB rises to 0, 68 N (point L3). Correspondingly to this, the static friction force at A increases from 0, 55 N to 0, 68 N (point K3).



During the automatic switch of A from rest to movement at a0 the force FSMA drops down to FKA D 0; 57 N (point K4). At that point, by way of the hands, the force at B decreases to FSB D 0; 57 N (point L4). From that instant on, sliding starts at A during which it moves from a0 to a1. At that, FKA rises to 0, 68 N (point K5). Simultaneously with it, FSB also rises to 0, 68 N (point L5). After that B again switches to a state of sliding, the force dropping to FKB D 0; 57 N (point L6) and so on.

The horizontal moves after L3 and K5 we use to symbolically represent the already-described automatic leaps that the stick does during switches.

These complex repeated changes of the forces up and down at the switching points happen along a vertical line. They are due to the already-described fact that the forces at the two supports are maintained as equal and thus the active change at one support is mirrored by a change at the other support. For a clearer representation of the processes, they are shown expanded on figure 7. On figure 6 these details are not given in order to present the process in its entirety.

These graphics, presented here for the first time, are extremely informative and allow the whole process to be understood clearly and in depth.

Acknowledgments

The authors would like to express great gratitude to the editor and referee, who were instrumental in making this paper significantly better than its initial version. We will also gladly consider any other appropriate ideas concerning the explanation of this very interesting experiment.

Received 13 March 2012, in final form 1 July 2012 doi:10.1088/0031-9120/47/6/701

References

[1] http://scienceblogs.com/dotphysics/2009/02/ friction-demo-with-a-meterstick.php www.physics.ucla.edu/demoweb/demomanual/ mechanics/torque/tricks with a meter stick. html www.lhup.edu/ dsimanek/scenario/demos.htm

Ehrlich R 1990 Turning the World Inside Out and 174 Other Simple Physics Demonstrations

(Princeton, NJ: Princeton University Press) pp 4950

Mancuso R V 1993 Quantitative analysis of moving two fingers under a meterstick Phys. Teach. 31 222

Massalha T et al 2011 Moving fingers under a stick: a laboratory activity Phys. Educ. 46 211 Ivanov D 1988 Eksperimentalni Zadachi Po Fizika

(Sofia: Narodna Prosveta) p 17, pp 746 (in Bulgarian)

Iona M 1993 But the meterstick has mass Phys. Teach. 31 327

Dragia T Ivanov graduated in engineering physics from Sofia University and then gained a PhD in physics didactics from St Petersburg, Russia in 1977. He has been a reader in physics didactics at Plovdiv University since 1980, becoming a professor in 2006. His areas of interest include physics didactics and multimedia teaching aids. He has written a number of books on practical physics experiments.

Stefan N Nikolov gained his BSc degree in engineering physics from Plovdiv University in 2005 and his MSc degree in medical and nuclear physics in 2007. He started his physics teaching experience as early as 2001 (aged 18) and has been teaching physics in one capacity or another ever since (still only as a part-time activity). He is currently employed at Plovdiv University as a physicist (lab technician) and is working towards obtaining his PhD in physics didactics. His areas of interest are teaching and communicating physics (and science in general) and he has given a number of one-off presentations on different physics topics on numerous occasions.


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