THE AIR TRACK (II) References: [] The Air Track (I) - First Year Physics Laoratory Manual (PHY38Y and PHYY) [] Berkeley Physics Laoratory, nd edition, McGraw-Hill Book Copany [3] E. Hecht: Physics: Calculus, nd edition, Brooks/Cole Introduction In Part of this second series of experients with the air track, you will investigate different types of collisions. The principles used to analyze the experiental data are: Newton's second and third laws which are used to develop the principle of conservation of oentu. In Part you will study a variety of dissipative (daping) forces. These forces include the various kinds of friction, agnetic daping, interaction forces in collisions which are not perfectly elastic, etc. Part 3 is dedicated to periodic otion. Haronic otion, daping and coupled oscillators are studied. In order to get weights, you have to do: Part (all) and. (Viscous daping) fro Part. To get 3 weights, you have to do Part (all), Part (all) and two experients fro Part 3.. Collisions. Theoretical ackground We have two gliders (asses and, velocities v and v, respectively) oving on a horizontal track. We take v to e positive if oves to the right and negative if to the left, and the sae for v. Both v and v are functions of tie. When the gliders collide, they stay in contact for a very short tie and they exert forces on each other. We assue that no other horizontal forces are exerted. Let the forces on and e F and F, respectively, with the sae sign conventions as for the velocities. According to Newton's second law: dv F dv F Newton's third law states that the two interaction forces have equal agnitude ut opposite directions: r r F -F (3) Coining (), () and (3) we get: () () or: d dv dv + ( v + v ) (4)
The quantity v is defined to e the oentu of the first glider, usually denoted y p, and siilarly for the other glider. Equation (4) states that the total oentu of the syste p +p does not change after the collision with respect to the initial value (its first derivative is zero). In all cases where there are no external forces, the total oentu of the colliding ojects is conserved. In an inelastic collision the final kinetic energy of the syste is different fro the initial one. All collisions etween acroscopic ojects are ore or less inelastic. In elastic collisions, the kinetic energy of the syste is conserved. We shall take the siple case of glider (velocity v ) colliding with glider at rest. After collision, velocities will e v and v, respectively. Let R e the ratio of final to initial kinetic energy of the syste: R v + v v (5) which ecoes, for the particular case of equal asses: R v + v v (6) Another useful quantity is the coefficient of restitution r: v - v r v The conservation of oentu equation can e written: + v v + v or: v v v (7) (8) By coining (8), (7), and (6), we get the siple result: ( r ) R + which shows that the final kinetic energy is the sae as the initial only if the collision is perfectly elastic (r ). (9)
A siilar expression ay e derived for the general case of unequal asses. It shows that: + r R + which proves again that kinetic energy is conserved when r.. Experient.. Equal asses Choose two identical gliders with upers. Place one glider at rest at aout the center of the track and direct the other toward it with a speed of ~c/s. Use the photogates on GATE function to easure the speeds of the gliders (efore and after the collision). In this arrangeent, the collision is to a good approxiation, an elastic one. To arrange an inelastic collision, attach a sall piece of Scotch tape or Velcro to each uper spring. In oth these situations, deterine the initial and final velocities, check the conservation of oentu and calculate the coefficient of restitution. Is kinetic energy conserved? What do you think it happens to the energy that is lost?.. Unequal asses Choose two unequal gliders and weigh the. Make qualitative oservations for oth > and <. Choose one particular coination for easureents. Note that in general it is necessary to easure three velocities: v and v can e easured with the two photogates; v with a stopwatch. Investigate: conservation of oentu and energy. Optional: investigate an inelastic collision with unequal asses...3 Action at a distance: the agnetic interaction force Sall ceraic agnets can e attached to the ends of the gliders, y using doule sided Scotch tape. Be sure that the agnets are oriented so that they repel each other, do not let the strike each other; they are ade fro a rittle aterial and are easily roken. Raise the track y a sall angle and arrange the gliders at the two ends. The glider near the raised end will coe to equiliriu at a position where the agnetic force just alances the force gsina down the track. Carefully easure the distance etween agnets. You ay repeat the easureent for several track angles a (ake sure to easure the elevation so that a can e calculated). Calculate the agnetic force at each position and plot a graph showing force as a function of distance..3 Questions on Part i) Suppose we could place a sall explosive charge on one of the upers, so that it detonates at the tie of collision, pushing the two gliders apart. Will the oentu still e conserved? Explain. Will kinetic energy e conserved? ii) What effect will the viscous friction of the supporting air layer have on your conclusions regarding conservation of oentu? () 3
. Dissipative forces. Viscous daping.. Theoretical ackground In this series of experients, you will study dissipative (daping) forces which act to dissipate echanical energy. Aong the, there are: friction, agnetic daping forces, interaction forces in collisions which are not perfectly elastic, etc. The principal source of frictions on the air track is the viscosity of the thin layer of air etween the glider and the track. The viscous friction force ay e written as: ηav F - d where η is a constant characteristic of the fluid, called viscosity, A is the surface area of the air layer and d is its thickness. The negative sign indicates that the direction of F is always opposite to that of the velocity. Force is proportional to velocity: F - v () where the constant depends on the properties of the air layer (viscosity, thickness, surface area). If a glider oves on a level track in the asence of any other forces except viscous forces, the equations of otion are: F a or - v dv showing that the rate of decrease of velocity is proportional to the velocity itself. To deterine the distance the glider travels efore stopping, we express dv/ in ters of dv/dx, using the chain rule for derivatives: () (3) dv By sustituting (3) in (4), we otain: which can e integrated to give: dv dx dx dv - dx v v - dv dx x v (4) (5) (6) 4
Equation (6) shows that a glider launched at an initial speed v coes at rest (v ) after traveling a distance: x v / (7) An interesting application of the dissipative forces is provided y the air track analogue to a ouncing all. If the track is tilted to an angle α and a glider is released fro rest at the top end, it will ounce at the otto end ut will not regain its original height. After a series of ounces, the glider eventually coes to rest at the otto of the tilted track. Since the gravitational force is conservative, the loss of energy is due entirely to the work done against the frictional force. The work done y friction during the first descent fro initial position x is: W - x Fdx - x (-v)dx Since the acceleration of the glider is approxiately a gsinα, the speed v at any position x is given y: (8) v ax gx sin α (9) sustituting the expression for v into (8) and integrating, we otain: (a) W - / (x 3 3 / ) The work done on the return trip is approxiately the sae, so the total change in energy due to viscous friction is W. If we take the potential energies at the ends of the first coplete trip: - initial: (gsinα)x - final: (gsinα)x, the corresponding loss of energy is: gsinα (x - x ) gsinα x By coining this result with Eq.() we otain: () x - (x 3(a) 3 / ) / () Thus if only the viscous forces are responsile for the energy loss, the change in height after the first ounce is proportional to the 3/ power of the original height, and so on. Energy can e lost also y collisions of upers to the end pieces of the track. The ratio of kinetic energies just after and just efore the ipact is r (r is the coefficient of restitution - see definition in Eq. 7). 5
.. Experients The track has to e carefully leveled. Launch a glider, easure its initial velocity and the total distance it travels efore it stops. For this easureent, the upers ay e considered perfectly elastic. Deterine the daping constant using eq. (7). Add weights to the glider to approxiately doule its ass and repeat the aove oservations and calculations. How does the value of change? Explain. For the "ouncing all" experient, tilt the track aout 5 illiradians. Release a glider fro the top; record its initial position and its axiu height after each ounce. Note that the position at the otto of the track ay not e at the zero point of the scale, in which case it ust e sutracted fro each reading. To analyze the ouncing all data, it is useful to take the logarith of oth sides in Eq. (): 3/ 3 log( - x) log x + log 3(a) / Thus, if the ouncing all ehaves according to this equation, the graph of log(- x) versus logx should e a straight line with a slope of 3/. If not only the viscous forces, ut also the collision to the end of the track uper deterines the energy loss, the slope of the graph should e etween and 3/.. Magnetic daping.. Theoretical ackground When an electrical conductor oves through a agnetic field, the changing agnetic flux in the conductor induces currents of agnitude proportional to the rate of change of flux and thus to the velocity. These eddy currents in turn experience a force which at each point is proportional to the field at that point. The direction of the force on the conductor is always such as to oppose the relative otion: F -'v (8) In this case, the constant ' is proportional to the electric conductivity of the aterial, to the area of the conductor over which the agnetic field extends and to the square of the agnetic field intensity... Experient Attach four agnets syetrically to the glider. Attach enough weight to another glider to give it the sae total ass as the glider with agnets. Place the two on the track and push the together (the agnetic glider in ack) to give the the sae initial velocity. Note that the agnetically daped glider lags increasingly ehind the other. Deterine the total daping constant for the agnetically daped glider y the sae ethod descried aove. Note that you are easuring the 'total ' due to oth viscous and agnetic daping. () 6
.3 Questions on Part i) For a glider with only viscous air daping, how does the daping constant vary with the ass of the glider? Why should this variation e expected? ii) When a glider on a tilted track is given an initial velocity v, show that if the track is sufficiently long, the glider will reach a final velocity (terinal velocity) which is independent of v. Derive an expression for the terinal velocity. iii) Is the effect of air surrounding the glider significant in coparison with the effect of the air layer etween glider and track, in deterining the total frictional force? Explain. Part 3 Periodic otion 3. Theoretical ackground When the force on a ody is proportional to the displaceent of the ody fro equiliriu and is directed toward the equiliriu position, there is a repetitive ackand-forth otion aout this position, called periodic otion. The oscillations of a ass on a spring, the otion of a pendulu, the virations of a stringed usical instruent, are failiar exaples of periodic otion. Let the displaceent of the ass fro equiliriu e x. The force is given y: F -kx (9) where k is a constant called the force constant for the syste. According to Newton's second law: d x - kx a It is easy to verify that functions: x x x x cosωt sin ωt () () are solutions of equation (), where x is a constant called the aplitude and ω (the angular frequency) is defined y: ω k Each tie the quantity ωt increases y π, the otion goes through one cycle. The tie for one cycle is called period (T): () π T π ω Ł k ł (3) 7
The nuer of cycles per unit tie is called frequency (f): ω πf f T π T Figure presents a odel, coprising of a glider placed on a horizontal air track and attached at its ends to identical springs. (4) k k Cord Air Track Figure The force constants of the two springs are equal to k. In order to stretch either spring y x, a force equal to k x has to e applied. The ass is displaced y distance x to the right of its equiliriu position. The force of the left spring increases y k x while that of the right one decreases y the sae aount. The result is a net force to the left with agnitude k x, so the force constant to e used in Eq. (9) and () is k k. The total energy of the syste fro Fig. is conserved. When the ass reaches the endpoints of its otion and stops, the energy is entirely potential energy; when it passes the equiliriu position, the energy is entirely kinetic energy. The average potential energy is equal to the average kinetic energy and each is equal to half the total energy. The presence of the daping forces in a real syste deterines a progressive decrease in the aplitude of the oscillations. The position of the ass is given y a ore coplicated function than (). Following the experients fro Part, we assue that the daping force is given y: F -v - dx where is the daping constant, characterizing the strength of the daping force. The rate at which oscillations die away depends on the agnitude of ; a large value of eans rapid decay, and the converse. By incorporating (5) is Newton's second law, we otain: (5) d x + dx + kx (6) 8
For the purpose of our experient, we shall use an approxiate analysis approach of the daped oscillations, ased on energy considerations. The energy lost during one cycle is given y the average rate of energy loss, calculated over that cycle. The rate of energy loss is given y (v)v v. To find the average value of v, we reeer that the average kinetic energy for a haronic oscillator is equal to its average potential energy and each of these quantities equals half the total energy E: v av E (7) The average rate of loss of energy is then: de av - v av - E (8) The tie required for one cycle is given y Eq. (3). The energy loss during one cycle is: π E - E -π Ł łł ω ł ( k) / E (9) Eq. (8) is a differential equation for E; its solution gives the energy as a function of tie: E E ( / )t e - where E is the initial total energy at tie t. The tie required for the energy to decrease to /e of its initial value is called relaxation tie and is given y (/). A useful constant that can e calculated now is called quality factor Q. It is defined as: Q πe E ω ( k) B / which eans π ties the ratio of axiu energy stored in the syste to the energy dissipated in one cycle. The aplitude is the ost directly oservale variale in an experient on periodic otion. It decreases with tie as: x x e - ( ( / )t ) / (3) (3) (3) 9
The exponential function aove is the square root of the function descriing the tie variation of E (E is proportional to x ). The relaxation tie for the aplitude of the oscillations can now e expressed as: τ The half-life T / is defined as the tie during which the aplitude drops to half its original value: (33) T / ln τln.386 (34) Eq. 33 can e used to give the quality factor Q: π T / π T/ ωτ Ł T łł ln ł ln T Q (35) 3. Experient 3.. Spring constant In order to copare the theoretical predictions with the oserved ehavior of the syste, the ass and spring constants ust e known. k Air Pulley Air Track Figure A suggested procedure for easuring the spring constant is shown in Fig.. The spring is attached etween the end of the track and a glider. Use a piece of agnetic recording tape (get it fro the Resource Center), tie it to the glider, pass it over the air pulley and attach a weight to the other end. Note the equiliriu position of the reference line on the glider. Add weights in g increents, up to g, recording the position of the reference line on the glider for each weight. Do not stretch the spring ore than c; eyond this it will e peranently defored. Plot extension of spring as a function of applied force (weight). Fro the graph, deterine the constant k.
3.. Siple haronic otion. To oserve siple haronic otion, reove the tape and attach a second spring as in Fig.. Attach a piece of cord to the end of this spring. Pull the cord enough to stretch each spring aout c and tie it to the end of the track. Displace the glider ~5 c fro its equiliriu position and release it. Tie cycles of the otion; find the period and the frequency. Repeat the easureent with saller and larger viration aplitudes. Record the aplitude for each trial. Is there any significant variation in frequency if larger or saller aplitudes are used? Fro the easured frequency and force constants, calculate the ass of the glider. 3..3 Daping Displace the glider 5 c fro the equiliriu and release it with no initial velocity. Count the nuer of cycles for the aplitude to decrease to half its original value. Calculate the quality factor of the syste, Q, and also the relaxation tie τ. Also copute the daping constant, copare with the value otained in Part. Add daping agnets to the glider and again deterine Q. Copare the result with q for a glider of the sae ass ut only viscous daping. Add ass (slotted weights) to the glider and oserve how Q changes with ass. Can you explain why Q varies? 3..4 Coupled oscillators Assele the syste as shown in Fig. 3: k k k Cord Air Track Figure 3 Pull the cord tight enough to extend each spring y ~ c. Displace one ass, holding the other fixed and release oth asses at once. Oserve the coplex nature of the otion. Displace oth asses toward the center y the sae aount and release the. Does the otion appear to e sinusoidal? This ode in which the asses ove in opposite directions is called the syetric ode. Measure the tie for oscillations, copute the period or frequency and copare with: ω Ł 3k ł / (see [], p.3) Displace the two asses in the sae direction y equal aounts and release the. Is the otion sinusoidal? Deterine again the frequency and copare with the theoretical prediction. This ode is called antisyetric.
3.3 Questions to Part 3 i) How is the otion of the syste shown in Fig. related to the otion of a siple pendulu? ii) If two identical springs, each with spring constant k, are connected in series, what is the resulting spring constant? What if they are connected in parallel? iii) In the aove analysis of haronic oscillators, the asses of the springs have een neglected. Will the effect of spring ass e to increase or decrease the frequency? Explain. RMS -