To eel a oce Chapte 7 Chapte 7: Static fiction, toque and static equilibium A. Review of foce vectos Between the eath and a small mass, gavitational foces of equal magnitude and opposite diection act on the masses. On a stationay mass, o on a mass maintaining a constant speed and diection nea the eath, additional foces must act to balance the eath's gavitational foce. A balanced foce condition can be vey useful in detemining the individual foces acting on a mass. A fomal pocedue has been pesented (see Equation 2.1) to detemine if the foce vectos acting on an object ae balanced. This pocedue is eviewed hee to pepae fo a discussion of static fiction and toque. oce vectos ae constucted by taking the foce magnitude (always a positive numbe) and attaching a sign: plus fo one diection o minus fo the opposite diection. The vecto quantity, containing both magnitude and diection, is witten symbolically as a bold oman lette, such as. The foces, 1 =+ and 2 =, ae vectos with a common magnitude, but point in opposite diections. It is entiely you own decision which diection you would like to choose as positive, but once you choose a convention you must stick with it. If the foces acting on a mass ae in balance, it means that the vecto sum of the foces (the net foce) is zeo. Balanced foces do not affect the motion of an object: the object will emain stationay o will continue with the same speed and diection. Balanced foces, howeve, will compess o stetch the object. If thee ae only two foces then the balance condition demands that the foces have equal magnitudes and opposite diections. The net foce on an object is the esult of the sum given in Equation 2.1: = + + + (7.1) net 1 2... n. This sum of the extenal foce vectos acting on an object is called the "net foce vecto" net. If thee ae only two foce vectos (n = 2) with equal magnitudes and opposite diections, 1 =+ and 2 =, then the foces ae in balance and the net foce acting on the object, detemined by substitution into Equation 7.1: net = 1 + 2 = ( + )+ ( ) = ( + ) ( + ) = 0. As expected, net is zeo. When net = ± net, whee net is the net foce magnitude (non- - zeo) and the sign (±) is the net foce diection, the motion of the object will be affected (its speed o diection will change). If we know, though some means (such as, the object is stationay o coasting along with a constant speed and diection), that foces acting on a mass ae in balance, then the net foce, given by Equation 7.1, can be set equal to zeo, net = 0. If, in 65
To eel a oce Chapte 7 addition, all but one of the foce vectos acting on the mass ae known, the value of the missing foce vecto can be detemined by solving Equation 7.1, with net = 0. If the speed o diection of an object changes (not stationay and not moving with a constant speed and diection) then the net foce acting on the object (Equation 7.1) will be non-zeo. The sign of the net foce vecto, net, is the diection of the net foce. The discussion of how to detemine, net, fo objects that ae acceleating (changing speed o diection) will be coveed in a late chapte. o now, to hone you skills in applying Equation 7.1, the examples povided in lectue and in homewok will have net = 0. B. Static iction To move a mass that is secued to the gound with scews, glue, Velco (locking sufaces), o othe constaint, equies a lage foce. Until the secuing agents yield o beak, they eact elastically, distoting sufficiently to geneate compession o tension foces that balance the applied foce. Even without a constaint secuing it to the gound, a mass will not begin to move until a hoizontal foce above a citical value is applied. If the mass does not begin to move in esponse to a hoizontal foce, a balancing foce (o foces) must also act on the mass. Static fiction is the common name given to this additional foce. Electomagnetic foces between the atoms of two objects in contact cause fictional foces as they attempt to slide on each othe. Static fiction behaves like a secuing agent: the magnitude of the static fictional foce acting on an object will be equal in magnitude and opposite in diection to the applied foce, and will theefoe pevent the motion of the object. The details of how this foce is geneated ae complex; howeve, the geneal chaacte of the fictional foce can be seen with a simple model epoducing its elastic behavio. A lage mass, M, stats with no hoizontal applied foce, and has a compession foce, C, at the inteface between it and the gound, with a magnitude equal to the weight of the mass, W = Mg, as shown in igue 7.1. The compession measues how had the two sufaces ae being pessed togethe. When pushed fom the side, a lage mass (same physical size) it is moe difficult to get moving because the compession foces ae lage. Also, it is usually moe difficult to get a M mass moving on a ough suface than on a C = Mg smooth one. The oughness of a suface is chaacteized by a numbe, µ (mu), that nomally takes values between 0 and 1 depending on the "stickiness" of the suface; µ = 0 fo a fictionless sufaces and, µ ~ 1, fo ough sufaces. igue 7.1 A mass, M, compesses the gound with a foce, C, equal to the weight of the mass, Mg. C 66
To eel a oce Chapte 7 A lage mass esting on the gound is pushed, as shown in igue 7.2, to the ight by a foce,, e.g., by a peson lying between the mass and wall, (fo claity, the compession foce vectos, C, shown in the pevious figue, ae not shown hee). The M f (on the mass) f (on the gound) see detail below igue 7.2 Two fictional foces, R, geneated by pushing on the mass and gound with a foce. mass emains stationay due to fictional foces, f, between the two sufaces. The fictional foce acting on the mass is equal in magnitude and opposite in diection to that of the applied foce,, and will incease in popotion to the applied foce until eaching a maximum value of, cit =µ C. If nothing else touches the mass, C = Mg, and theefoe, cit =µ Mg. The mass begins to slide when is geate than the citical fictional foce, > cit, cit =µ Mg (7.2) The citical hoizontal foce, cit, shift depends on the magnitude of a vetical foce, Mass Mg. This may seem quite odd, but it does make sense in a model inceasing the numbe of active atoms on moe compessed sufaces. Geneation of static fictional foces at the mating sufaces Gound of the mass and the gound can be modeled igue 7.3 Detail of the mating sufaces of the Mass and the Gound. using idealized elements, as shown in igue 7.3. Pojections fom the suface of the mass and gound epesent oughness at the atomic scale. A natual length fiction sping attached to these pojections simulates the electomagnetic foces between the atoms. Mass When the mass is pushed to the ight the T mass will move vey slightly to the ight, thus stetching the micoscopic fiction spings, as T shown in igue 7.4 (pushing to the left Gound 123 x compesses the sping and ceates compession igue 7.4 Detail of mating sufaces when the foces pushing back on both objects). The mass is pushed to the ight. tension foces (T) in the stetched sping attempt to pull the top mass towad the left and the gound to the ight. The fictional foces on the mass and gound ae simulated in this model by the tension foces of a lage numbe of spings. 67
To eel a oce Chapte 7 The magnitude of the tension foces of the fiction sping will be equal to the foce applied to the mass by the pushe as shown in igue 7.2. The foce of the peson against the wall and the fictional foce acting on the gound ae also equal in magnitude and opposite in diection). The net effect of the applied and fictional foces is to keep the mass in place, compessing it between the applied foce and the fictional foce on the mass. The peson tied to push the mass, and instead, compesses the mass and stetches the gound. It is often said that it is unnecessay to discuss the oigin of fictional foces. Once thee is a fictional foce on the mass, thee must be anothe foce on the gound with equal magnitude and opposite diection by Newton s 3 d Law. Nevetheless should be comfoting to know that an explanation of Newton s 3 d Law in each application is possible using the same model of elastic foces. Extending the model can epoduce othe popeties of the fictional foce. The mass will begin to slide if a foce is applied to the mass that stetches (o compesses) the spings too fa and they beak, o the pojections holding the spings beak. Inceasing the foce that is squeezing the two sufaces togethe will incease the numbe of spings that ae active, and inceases the foce necessay to make the mass stat to slide, in ageement with the behavio of the eal fictional foce. Some aspects of the fictional foce ae not epoduced, but modeling static fiction with the elastic foces of tension and compession, is easonably accuate. Once a mass is pushed with a foce geate than cit, it begins to slide. While sliding a new foce, called sliding fiction, will begin to affect the motion. The foce of sliding fiction gows with speed, so that at some speed the sliding fictional foce will balance the applied foce, and the motion will then have a constant speed and diection. 68
To eel a oce Chapte 7 C. Toque and Twisting (Sections C and D ae optional eading) In ealie chaptes, examples wee caefully chosen such that the foces wee symmetically placed: fo evey foce on one side of a line though the cente C of the object thee was anothe foce on the othe side with the same magnitude, diection, and distance fom the line. Anothe symmetical case is shown in igue 7.5 Pushing on a dooknob. igue 7.5, whee foces, labeled, ae applied to the uppe and lowe suface of a dooknob in the same diection (suely not the way to tun a dooknob). The two foces push the dooknob to the ight and, not supisingly, nothing moves o tuns. The doo (hinges on the ight) is compessed and geneates a compession foce, C = 2, acting on the dooknob, balancing the applied foces. The example shown in igue 7.6, howeve, diffes fom those consideed peviously. What makes this situation diffeent is the twisting action, called "toque", that tuns this dooknob clockwise. The dooknob will not move left o ight because the two foces have equal magnitudes and opposite diections, and theefoe, ae in balance. The dooknob tuns igue 7.6 oces that will howeve, because the toques do not balance! tun a dooknob Each of the foces acting on an object will geneate an associated toque. The Geek lette Tau () is the symbol used fo toque (T is aleady used fo tension). A foce can be applied at a point on a dooknob, as shown in igue 7.7, at angles, θ, fom 0 to 90, with espect to a line, of length, fom that point to the cente of the dooknob. If a igue 7.7 A foce applied at diffeent angles to a dooknob. foce is applied at θ =90 o, shown on the left, then the magnitude of the toque,, is a maximum given by: θ=90 o θ θ=0 o =, the poduct of the magnitude of the foce and the distance to the pivot. When the foce is diected towad the pivot point, as shown on the ight, the toque is zeo. The toque is less than the maximum fo intemediate angles shown in the cente. In this text, to simply calculations, a foce will be applied to an object only at the angle 0, whee the toque is (7.3) 69
To eel a oce Chapte 7 zeo, o at 90, whee the toque is given by Equation 7.3. Note that a foce can be applied without causing a toque but a toque cannot be ceated without a foce. The foces, shown in igue 7.8, ae in opposite diections but they cause toques that ae in the same diection (tun the dooknob clockwise). oces applied in opposite diections, and on opposite sides of the dooknob, geneate toques in the same diection. Thee is a net toque but at the same time the foces ae igue 7.8 Geomety of a Dooknob balanced (applied in opposite diections) so that the object will not stat to move to eithe side. Toque is a vecto quantity, epesented as a bold lette Tau (), and the magnitude and diection must be specified with a = = + pocedue simila to the one used fo foces. A positive toque will twist in the clockwise diection (most students pefe clockwise as the positive diection), while a negative toque will igue 7.9 Toque vectos twist in the counte clockwise diection. It is unusual to daw toque vectos, but I will use a cuved aow in the diection of the toque, labeled by the magnitude of the toque; a positive toque vecto, shown in igue 7.9, is on the ight, while a negative toque vecto is on the left. Each of the foces acting on the dooknob will geneate an associated toque. As shown on the left of igue 7.10, the toque vectos, =+ (uppe), and, = (lowe) ae geneated by foces pointing to the ight, yielding a net toque, net = 0 (the toques balance). A pai of balanced toques does not tun the knob, howeve the unbalanced foces will push the dooknob to the ight. oces that cause toques in the same diection, as shown on the ight in igue 7.10, geneate a net toque, net =+2, and will tun the dooknob clockwise. At the same time, the foce vectos will balance so that the net foce is zeo. Theefoe, as desied, the dooknob will tun and will not be pushed lateally in eithe diection. igue 7.10 Toque vectos geneated by foces acting on a doo knob 70
To eel a oce Chapte 7 D. Genealized static equilibium a) b) The sum of the toques acting 1 1 1 on an object is called the net toque, 1 and if non-zeo, the toques ae pivot pivot unbalanced, and a non-otating object will begin to tun in the diection of 2 the net toque. An example, shown in 2 2 igue 7.11a, whee equal foces ae 2 applied at diffeent distances, 2 igue 7.11 oces causing a non-zeo net toque on a ba. 1, fom a fixed pivot, esults in a negative net toque (non-zeo), and that object begins to tun counteclockwise. Anothe example, shown in igue 7.11b, whee foces have diffeent magnitudes, 1 2, but ae applied at the same distance fom the pivot, esults in a positive net toque, and that object begins to tun clockwise. With asymmeties in both foce and distance, whee the lage foce is applied at a smalle distance fom the pivot, as shown in igue 7.12, 1 1 a zeo net toque can esult. Deived below, is the condition on the two foces acting pependicula to a ba 1 pivot and pivot that esult in otational equilibium: 1+ 2 = ( + 1 1)+ ( 2 2)= 0 2 (Eq. 7.4) 11= 22 2 2 o an object to be in static equilibium (not begin igue 7.12 oces causing a to move lateally o otate), no net foce and no net toque can act on the object. These two conditions must be met simultaneously and can be expessed as: zeo net toque on a ba. = + + + = net 1 2... n 0, net = 1 + 2 +... + n = 0, (Eq. 7.5) whee the toque vectos, i, have the magnitude, i = i i fo each foce (i = 1, n) acting pependicula to a line fom the pivot to the point of action. At the fixed (stationay) pivots of the bas shown in igues 7.11 and 7.12, a foce must act to the left to balance the foces shown acting to the ight. The foce acting on the pivot acts at zeo distance fom the pivot and does not geneate any toque. In the solution of a poblem involving static equilibium, this featue of toque povides a (hidden) constaint, and the balanced condition on toque should be applied fist. 71
To eel a oce Chapte 7 o homewok poblems whee a given set of masses is attached to a ba in static equilibium, the application of Equations 7.5, can esolve all the foces and toques acting on the object. If one (o both) of the equations is not satisfied (not zeo) then a stationay object must begin to move in the diection of the net foce, o begin to tun in the diection of the net toque, o (if both ae non-zeo), begin to move and tun. 72
To eel a oce Chapte 7 Chapte Summay: An stationay mass o one moving with a constant speed and diection will have no net foce, net = 0, acting on it. A mass with a changing speed o diection has a net foce acting on it, net 0. On a stationay mass, a foce applied paallel to the hoizontal mating sufaces, geneates a balancing static fictional foce, f, pushing back on the mass. The static fictional foce will immediately become zeo and the object will begin to slide if the magnitude of the applied foce is lage than a citical value, cit. The citical foce to slide is given by, cit = µc, whee µ is the coefficient of fiction and C is the compession foce between the sufaces in contact. Toque and foce ae sepaate quantities with diffeent units. You cannot add a toque to a foce. The magnitude of the toque,, caused by a foce applied at 90 to a line of length dawn fom the pivot to the point the foce is applied is =. The magnitude of the toque is zeo fo a foce applied at 0 to a line dawn fom the pivot to the point the foce is applied. The diection of a toque vecto,, is positive if it tends to otate an object clockwise and negative if it tends to otate an object counte-clockwise. o static equilibium, the geneal conditions on the foces applied to an object ae: net = 1 + 2 +... + n = 0, and net = 1 + 2 +... + n = 0, whee i = i i fo foces applied at 90 to a line of length dawn fom the pivot to the point whee the foce is applied, and i = 0 fo foces applied at 0 to the line. 73