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Rolling Friction on a Wheeled Laboratory Cart

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Rolling Friction on a Wheeled Laboratory Cart P APERS www.iop.org/journals/physed Rolling friction on a wheeled laboratory cart Carl E Mungan Physics Department, US Naval Academy, Annapolis, MD 21402-1363, USA E-mail: [email protected] Abstract A simple model is developed that predicts the coefficient of rolling friction for an undriven laboratory cart on a track that is approximately independent of the mass loaded onto the cart and of the angle of inclination of the track. The model includes both deformation of the wheels/track and frictional torque at the axles/bearings. The concept of rolling friction is contrasted with the static or kinetic friction that in general is also present, such as for a cylinder or ball rolling along a horizontal or inclined surface. Introduction Part of the confusion is understandable, Small rolling carts on metal tracks have become in that static or kinetic friction, under many common in introductory physics laboratories at circumstances, dominates and hence masks the universities and secondary schools. Examples rolling friction, which is always present and include PASCO’s ‘Collision Cart’, Vernier’s distinct from either of these other two kinds of ‘Standard Cart’, PHYWE’s ‘Low Friction Cart’, friction for a rolling object. For example, if a string and PSSC’s ‘Dynamics Cart’. Nevertheless, is wound around the axle of a rolling cylinder [7] there exists confusion about how to simply and or of a spool [8] and used to drive its motion, then accurately model the resistance that such carts static friction usually arises to prevent the object encounter when they move on level or inclined from slipping. (Depending on how it is pulled, tracks. static friction is not always needed [9], but usually For example, two articles [1, 2] refer to it is.) However, static friction is not necessarily a the friction experienced by the carts as ‘kinetic’, ‘resistance’ to motion: in fact, it can point in (as which is incorrect because they roll not slide opposed to against) the direction of motion [10]. along the track. A more recent paper [3] does In any case it does not dissipate any mechanical not specifically use the adjective ‘kinetic’, but energy because its point of contact at the bottom of nevertheless subscripts the coefficient of friction the rolling object has zero instantaneous velocity. between the cart and track with the letter ‘k’which (Zero displacement means zero work done. It is presumably is an abbreviation for that adjective. static after all.) Similarly, a cylinder rolling up or Other authors hedge their bets by simply referring down an incline is driven by gravity, and a vehicle to the force opposing the motion of the carts as (such as a car or bicycle) accelerating along a level ‘friction’ without a qualifying adjective [4, 5]and road is driven by a motor or by pedalling. In such the coefficient of friction by the unsubscripted cases, static friction again dominates (assuming symbol ‘μ’[6], even though most introductory the rolling is without slipping) and can point either physics textbooks only denote friction as either forward or backward relative to the direction of static or kinetic. motion [8]. If slipping does occur, then kinetic 288 P HYSICS E DUCATION 47 (3) 0031-9120/12/030288+05$33.00 © 2012 IOP Publishing Ltd Rolling friction on a wheeled laboratory cart N υ ω τ α a τ ƒ O θ mg N θ ~ R ƒ C Figure 1. Free-body diagram for a cart rolling up an of the cart points uphill, but since the cart is slowing D down, its acceleration a points downhill). The normal force N, frictional force f and axle torque Figure 2. Enlarged view of the contact forces and axle torque at the wheels. This diagram has been four wheels. The axle has been labelled as the origin O and the effective point of contact of the forces between the wheel and track as C, separated horizontally from each other by a distance D. The flattening of the friction will dominate, as is purposely exploited in wheel has been exaggerated for clarity. The some games such as bowling and billiards [11]. a In contrast, if a cylinder or ball is freely in figure 1. rolled (i.e. without being driven by a string or motor) along a horizontal surface, it will depression in (or bowing of) the track out of which eventually come to a stop (even if it does not it has to continuously roll. strike any small bits of dirt or other obstacles). Consider a cart of mass m that is free rolling This slowing will arise even in the absence of up an incline, as sketched in figure 1. The total air drag, as one can verify by rolling the object frictional force f on the cart from the track is inside an evacuated bell jar. In the absence of assumed to oppose the direction of motion υ of slipping, the resistive force cannot be ordinary the cart. Similarly, the total frictional torque τ at static friction, as explained in figure 1 of [12]; for the four bearings opposes the direction of rotation example, if the friction force pointed backward ω of the wheels. Force components perpendicular (to translationally decelerate the object), then it to the incline must balance so that would simultaneously rotationally accelerate the cylinder about its centre, which cannot be right! N = mg cos θ (1) The object could partially stick to the surface and have to be ‘peeled’ away from it and thereby get where N is the sum of the normal forces at all τ slowed down [13], but that cannot be a complete four wheels. (In general, N, f and will not explanation, as nonadhesive rolling objects also be equally divided among the four wheels.) Force slow down. A simple but comprehensive model components along the incline imply of rolling friction is developed in this article. mg sin θ + f = ma. (2) General theory of rolling friction The rotational form of Newton’s second law about point O in the enlarged view in figure 2 is Rolling friction on the wheels of a cart can be modelled using the ideas of Krasner [12]. There ND+ τ − fR= Iα (3) are two contributions to the friction: deformation of the wheels and/or track, and the torque at the where the angular acceleration of the wheels is axles (or bearings) opposing the rotation of the α = a/R assuming they do not slip, and the wheels. The deformation will be represented here sum of the moments of inertia of the four wheels as a flattening of the wheels, although in general is I = γ MR2.HereM is the total mass it can also arise from the cart creating a slight of the four wheels (each of radius R)andγ is May 2012 P HYSICS E DUCATION 289 C E Mungan a mass distribution factor, of the order of 0.5 (1) Consider a single round object (so that if the wheels are approximately uniform discs. M = m) rolling on a horizontal surface (so that Solving equation (2)fora and substituting it into θ = 0). Notice that the term proportional to k in equation (3) leads to equation (7) then drops out, as expected since there is no axle. One finds that μ = (D/R)/(1 + γ), ND+ τ − fR= γ MR(g sin θ + f/m). (4) which is the coefficient of rolling friction (usually written as μ ) that explains why the object will Now suppose we can relate the frictional force r eventually roll to a stop [17]. This coefficient on the cart to the normal force on the entire cart agrees with Krasner’s equation (5) after correcting via a coefficient of friction μ (labelled without the typo whereby he multiplied (D/R) and (1+γ) subscripts for the moment), instead of dividing them [12]. f = μN = μmgcos θ (5) (2) Again consider a single round object (so that M = m) but this time rolling on using equation (1) to get the second equality. an incline. Suppose the object and track are Similarly suppose that the frictional torque on the sufficiently rigid that they cannot deform (so that axles can be related to the normal component of D = 0). Now equation (7) simplifies to μ = − the gravitational load on them from the body of − tan θ/(1 + γ 1). The minus sign implies that the cart without the wheels, (m − M)g cos θ,so the frictional force is directed up rather than down that the incline. This upward direction is necessary so τ = kr(m − M)g cos θ (6) that friction will angularly decelerate the object as it rolls up the ramp; the friction in this case where r is the radius of the axles and k is a is static not rolling [8]. We see that μ is not a (dimensionless) coefficient of frictional torque. It constant independent of θ, but instead its value is assumed that μ and k are independent of speed. automatically adjusts (up to a maximum value In support of that assumption, measurements of of μ if slipping is not to occur). It is more a ball on an incline [14] indicate that μ has less s conventional to write fs = μN =−mg sin θ/(1+ than a 10% dependence on speed for angular − γ 1) from equation (5) and not refer directly to velocities up to 1.5 rad s−1. Given that the μ at all. Substituting this expression for f into wheels of a PASCO cart have a diameter of s equation (2) leads to the familiar result a = 2R = 2.54 ± 0.01 cm, this limiting angular g sin θ/(1 + γ) for the translational acceleration velocity requires that the cart travels slower than of a rolling object on an inclined plane.
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