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130-420: Analogous with sliding friction , rolling resistance is often expressed as a coefficient times the normal force. This coefficient of rolling resistance is generally much smaller than the coefficient of sliding friction. Any coasting wheeled vehicle will gradually slow down due to rolling resistance including that of the bearings, but a train car with steel wheels running on steel rails will roll farther than

260-453: A bus of the same mass with rubber tires running on tarmac/asphalt . Factors that contribute to rolling resistance are the (amount of) deformation of the wheels, the deformation of the roadbed surface, and movement below the surface. Additional contributing factors include wheel diameter , load on wheel , surface adhesion, sliding, and relative micro-sliding between the surfaces of contact. The losses due to hysteresis also depend strongly on

390-430: A rail ). For railroads, this is called curve resistance but for roads it has (at least once) been called rolling resistance due to cornering . Rolling friction generates sound (vibrational) energy, as mechanical energy is converted to this form of energy due to the friction. One of the most common examples of rolling friction is the movement of motor vehicle tires on a roadway , a process which generates sound as

520-403: A tractive force equal to 70% of the maximum traction, slip resistance becomes 10 times larger than the basic rolling resistance. In order to apply any traction to the wheels, some slippage of the wheel is required. For trains climbing up a grade, this slip is normally 1.5% to 2.5%. Slip (also known as creep ) is normally roughly directly proportional to tractive effort . An exception is if

650-449: A 20% increase in load decreases Crr by 3%. But, if the inflation pressure is not changed, then a 20% increase in load results in a 4% increase in Crr. Of course, this will increase the rolling resistance by 20% due to the increase in load plus 1.2 x 4% due to the increase in Crr resulting in a 24.8% increase in rolling resistance. When a vehicle ( motor vehicle or railroad train ) goes around

780-433: A 5% slip can translate into a 200% increase in rolling resistance. This is partly because the tractive force applied during this slip is many times greater than the rolling resistance force and thus much more power per unit velocity is being applied (recall power = force x velocity so that power per unit of velocity is just force). So just a small percentage increase in circumferential velocity due to slip can translate into

910-432: A Crr of 0.00013 (axle load of 21 tonnes). For empty freight cars with axle loads of 5.5 tonnes, Crr goes up to 0.00020 at 60 km/h but at a low speed of 20 km/h it increases to 0.00024 and at a high speed (for freight trains) of 120 km/h it is 0.00028. The Crr obtained above is added to the Crr of the other components to obtain the total Crr for the wheels. The rolling resistance of steel wheels on steel rail of

1040-399: A by-product. The sound generated by automobile and truck tires as they roll (especially noticeable at highway speeds) is mostly due to the percussion of the tire treads, and compression (and subsequent decompression) of air temporarily captured within the treads. Several factors affect the magnitude of rolling resistance a tire generates: In a broad sense rolling resistance can be defined as

1170-549: A car of 1000 kg on asphalt will need a force of around 100  newtons for rolling (1000 kg × 9.81 m/s × 0.01 = 98.1 N). According to Dupuit (1837), rolling resistance (of wheeled carriages with wooden wheels with iron tires) is approximately inversely proportional to the square root of wheel diameter. This rule has been experimentally verified for cast iron wheels (8″ - 24″ diameter) on steel rail and for 19th century carriage wheels. But there are other tests on carriage wheels that do not agree. Theory of

1300-484: A coefficient (ratio)or a multiple thereof. If using pounds or kilograms as force units, mass is equal to weight (in earth's gravity a kilogram a mass weighs a kilogram and exerts a kilogram of force) so one could claim that C r r {\displaystyle C_{rr}} is also the force per unit mass in such units. The SI system would use N/tonne (N/T, N/t), which is 1000 g C r r {\displaystyle 1000gC_{rr}} and

1430-404: A curve, rolling resistance usually increases. If the curve is not banked so as to exactly counter the centrifugal force with an equal and opposing centripetal force due to the banking, then there will be a net unbalanced sideways force on the vehicle. This will result in increased rolling resistance. Banking is also known as "superelevation" or "cant" (not to be confused with rail cant of

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1560-407: A cylinder rolling on an elastic roadway also gives this same rule These contradict earlier (1785) tests by Coulomb of rolling wooden cylinders where Coulomb reported that rolling resistance was inversely proportional to the diameter of the wheel (known as "Coulomb's law"). This disputed (or wrongly applied) -"Coulomb's law" is still found in handbooks, however. For pneumatic tires on hard pavement, it

1690-415: A deformable material such that the energy of deformation is greater than the energy of recovery. The rubber compound in a tire exhibits hysteresis. As the tire rotates under the weight of the vehicle, it experiences repeated cycles of deformation and recovery, and it dissipates the hysteresis energy loss as heat. Hysteresis is the main cause of energy loss associated with rolling resistance and is attributed to

1820-434: A faster rate as the torque becomes higher. The rolling resistance coefficient, Crr, significantly decreases as the weight of the rail car per wheel increases. For example, an empty freight car had about twice the Crr as a loaded car (Crr=0.002 vs. Crr=0.001). This same "economy of scale" shows up in testing of mine rail cars. The theoretical Crr for a rigid wheel rolling on an elastic roadbed shows Crr inversely proportional to

1950-399: A force F A acting on a point that moves with velocity v A and the output power be a force F B acts on a point that moves with velocity v B . If there are no losses in the system, then P = F B v B = F A v A , {\displaystyle P=F_{\text{B}}v_{\text{B}}=F_{\text{A}}v_{\text{A}},} and

2080-645: A graphene sheet in the presence of graphene-adsorbed oxygen. Despite being a simplified model of friction, the Coulomb model is useful in many numerical simulation applications such as multibody systems and granular material . Even its most simple expression encapsulates the fundamental effects of sticking and sliding which are required in many applied cases, although specific algorithms have to be designed in order to efficiently numerically integrate mechanical systems with Coulomb friction and bilateral or unilateral contact. Some quite nonlinear effects , such as

2210-485: A logarithmic measure relative to a reference of 1 milliwatt, calories per hour, BTU per hour (BTU/h), and tons of refrigeration . As a simple example, burning one kilogram of coal releases more energy than detonating a kilogram of TNT , but because the TNT reaction releases energy more quickly, it delivers more power than the coal. If Δ W is the amount of work performed during a period of time of duration Δ t ,

2340-417: A loss of traction power which may even exceed the power loss due to basic (ordinary) rolling resistance. For railroads, this effect may be even more pronounced due to the low rolling resistance of steel wheels. It is shown that for a passenger car, when the tractive force is about 40% of the maximum traction, the slip resistance is almost equal to the basic rolling resistance (hysteresis loss). But in case of

2470-411: A new surface forms at the back of a sliding true contact, and existing surface disappears at the front of it. Since all surfaces involve the thermodynamic surface energy, work must be spent in creating the new surface, and energy is released as heat in removing the surface. Thus, a force is required to move the back of the contact, and frictional heat is released at the front. For certain applications, it

2600-400: A parameter describing the scaling behavior of surface asperities, is known to play an important role in determining the magnitude of the static friction. For surfaces in relative motion μ = μ k {\displaystyle \mu =\mu _{\mathrm {k} }} , where μ k {\displaystyle \mu _{\mathrm {k} }} is

2730-420: A periodic function of period T {\displaystyle T} . The peak power is simply defined by: P 0 = max [ p ( t ) ] . {\displaystyle P_{0}=\max[p(t)].} The peak power is not always readily measurable, however, and the measurement of the average power P a v g {\displaystyle P_{\mathrm {avg} }}

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2860-472: A rough body driven over a rough surface, the mechanical work done by the driver exceeds the mechanical work received by the surface. The lost work is accounted for by heat generated by friction. Over the years, for example in his 1879 thesis, but particularly in 1926, Planck advocated regarding the generation of heat by rubbing as the most specific way to define heat, and the prime example of an irreversible thermodynamic process. The focus of research during

2990-515: A single study has demonstrated the potential for an effectively negative coefficient of friction in the low-load regime , meaning that a decrease in normal force leads to an increase in friction. This contradicts everyday experience in which an increase in normal force leads to an increase in friction. This was reported in the journal Nature in October 2012 and involved the friction encountered by an atomic force microscope stylus when dragged across

3120-880: A slow rigid wheel on a perfectly elastic surface, not adjusted for velocity, can be calculated by C r r = z / d {\displaystyle C_{rr}={\sqrt {z/d}}} where The empirical formula for C r r {\displaystyle C_{rr}} for cast iron mine car wheels on steel rails is: C r r = 0.0048 ( 18 / D ) 1 2 ( 100 / W ) 1 4 = 0.0643988 W D 2 4 {\displaystyle C_{rr}=0.0048(18/D)^{\frac {1}{2}}(100/W)^{\frac {1}{4}}={\frac {0.0643988}{\sqrt[{4}]{WD^{2}}}}} where As an alternative to using C r r {\displaystyle C_{rr}} one can use b {\displaystyle b} , which

3250-419: A threshold value for this force, above which motion would commence. This maximum force is known as traction . The force of friction is always exerted in a direction that opposes movement (for kinetic friction) or potential movement (for static friction) between the two surfaces. For example, a curling stone sliding along the ice experiences a kinetic force slowing it down. For an example of potential movement,

3380-401: A ton of freight about 130 miles per gallon of fuel, indicating trains are more efficient overall. Sliding friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of the processes involved

3510-400: A tonne. This lighter weight per passenger, combined with the lower rolling resistance of steel wheels on steel rail means that an N700 Shinkansen is much more energy efficient than a typical automobile. In the case of freight, CSX ran an advertisement campaign in 2013 claiming that their freight trains move "a ton of freight 436 miles on a gallon of fuel", whereas some sources claim trucks move

3640-407: A train is far less than that of the rubber tires wheels of an automobile or truck. The weight of trains varies greatly; in some cases they may be much heavier per passenger or per net ton of freight than an automobile or truck, but in other cases they may be much lighter. As an example of a very heavy passenger train, in 1975, Amtrak passenger trains weighed a little over 7 tonnes per passenger, which

3770-434: A very poor approximation (for example, adhesive tape resists sliding even when there is no normal force, or a negative normal force). In this case, the frictional force may depend strongly on the area of contact. Some drag racing tires are adhesive for this reason. However, despite the complexity of the fundamental physics behind friction, the relationships are accurate enough to be useful in many applications. As of 2012 ,

3900-409: Is (highly ordered pyrolytic) graphite which can have a friction coefficient below 0.01. This ultralow-friction regime is called superlubricity . Static friction is friction between two or more solid objects that are not moving relative to each other. For example, static friction can prevent an object from sliding down a sloped surface. The coefficient of static friction, typically denoted as μ s ,

4030-498: Is a dimensionless scalar value which equals the ratio of the force of friction between two bodies and the force pressing them together, either during or at the onset of slipping. The coefficient of friction depends on the materials used; for example, ice on steel has a low coefficient of friction, while rubber on pavement has a high coefficient of friction. Coefficients of friction range from near zero to greater than one. The coefficient of friction between two surfaces of similar metals

Rolling resistance - Misplaced Pages Continue

4160-551: Is a different rolling resistance coefficient or coefficient of rolling friction with dimension of length. It is defined by the following formula: F = N b r {\displaystyle F={\frac {Nb}{r}}} where The above equation, where resistance is inversely proportional to radius r {\displaystyle r} seems to be based on the discredited "Coulomb's law" (Neither Coulomb's inverse square law nor Coulomb's law of friction). See dependence on diameter . Equating this equation with

4290-408: Is an empirical measurement   —   it has to be measured experimentally , and cannot be found through calculations. Rougher surfaces tend to have higher effective values. Both static and kinetic coefficients of friction depend on the pair of surfaces in contact; for a given pair of surfaces, the coefficient of static friction is usually larger than that of kinetic friction; in some sets

4420-495: Is an approximate model used to calculate the force of dry friction. It is governed by the model: F f ≤ μ F n , {\displaystyle F_{\mathrm {f} }\leq \mu F_{\mathrm {n} },} where The Coulomb friction F f {\displaystyle F_{\mathrm {f} }} may take any value from zero up to μ F n {\displaystyle \mu F_{\mathrm {n} }} , and

4550-413: Is assumed that all wheels are the same and bear identical weight. Thus:   C r r = 0.01 {\displaystyle \ C_{rr}=0.01} means that it would only take 0.01 pounds to tow a vehicle weighing one pound. For a 1000-pound vehicle, it would take 1000 times more tow force, i.e. 10 pounds. One could say that C r r {\displaystyle C_{rr}}

4680-698: Is asymmetrical and is shifted to the right. The line of action of the (aggregate) vertical force no longer passes through the centers of the cylinders. This means that a moment occurs that tends to retard the rolling motion. Materials that have a large hysteresis effect, such as rubber, which bounce back slowly, exhibit more rolling resistance than materials with a small hysteresis effect that bounce back more quickly and more completely, such as steel or silica . Low rolling resistance tires typically incorporate silica in place of carbon black in their tread compounds to reduce low-frequency hysteresis without compromising traction. Note that railroads also have hysteresis in

4810-414: Is called tribology , and has a history of more than 2000 years. Friction can have dramatic consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire . Another important consequence of many types of friction can be wear , which may lead to performance degradation or damage to components. It is known that frictional energy losses account for about 20% of

4940-412: Is constant, the amount of work performed in time period t can be calculated as W = P t . {\displaystyle W=Pt.} In the context of energy conversion, it is more customary to use the symbol E rather than W . Power in mechanical systems is the combination of forces and movement. In particular, power is the product of a force on an object and the object's velocity, or

5070-432: Is done in this article. They just sum up all the resistance forces (including aerodynamic drag) and call the sum basic train resistance (or the like). Since railroad rolling resistance in the broad sense may be a few times larger than just the pure rolling resistance reported values may be in serious conflict since they may be based on different definitions of "rolling resistance". The train's engines must, of course, provide

5200-528: Is force per unit mass, where g is the acceleration of gravity in SI units (meters per second square). The above shows resistance proportional to C r r {\displaystyle C_{rr}} but does not explicitly show any variation with speed, loads , torque , surface roughness, diameter , tire inflation/wear, etc., because C r r {\displaystyle C_{rr}} itself varies with those factors. It might seem from

5330-404: Is given by P ( t ) = p Q , {\displaystyle P(t)=pQ,} where p is pressure in pascals or N/m , and Q is volumetric flow rate in m /s in SI units. If a mechanical system has no losses, then the input power must equal the output power. This provides a simple formula for the mechanical advantage of the system. Let the input power to a device be

Rolling resistance - Misplaced Pages Continue

5460-437: Is greater than that between two surfaces of different metals; for example, brass has a higher coefficient of friction when moved against brass, but less if moved against steel or aluminum. For surfaces at rest relative to each other, μ = μ s {\displaystyle \mu =\mu _{\mathrm {s} }} , where μ s {\displaystyle \mu _{\mathrm {s} }}

5590-424: Is impending, is sometimes referred to as limiting friction , although this term is not used universally. Kinetic friction , also known as dynamic friction or sliding friction , occurs when two objects are moving relative to each other and rub together (like a sled on the ground). The coefficient of kinetic friction is typically denoted as μ k , and is usually less than the coefficient of static friction for

5720-420: Is in lb(tow-force)/lb(vehicle weight). Since this lb/lb is force divided by force, C r r {\displaystyle C_{rr}} is dimensionless. Multiply it by 100 and you get the percent (%) of the weight of the vehicle required to maintain slow steady speed. C r r {\displaystyle C_{rr}} is often multiplied by 1000 to get the parts per thousand, which

5850-433: Is in part due to the fact that there is some slipping of the wheel, and for pneumatic tires, there is more flexing of the sidewalls due to the torque. Slip is defined such that a 2% slip means that the circumferential speed of the driving wheel exceeds the speed of the vehicle by 2%. A small percentage slip can result in a slip resistance which is much larger than the basic rolling resistance. For example, for pneumatic tires,

5980-427: Is largely dependent on the tractive force , coefficient of friction, normal load, etc. "Applied torque" may either be driving torque applied by a motor (often through a transmission ) or a braking torque applied by brakes (including regenerative braking ). Such torques results in energy dissipation (above that due to the basic rolling resistance of a freely rolling, i.e. except slip resistance). This additional loss

6110-415: Is maintained that μ is always < 1, but this is not true. While in most relevant applications μ < 1, a value above 1 merely implies that the force required to slide an object along the surface is greater than the normal force of the surface on the object. For example, silicone rubber or acrylic rubber -coated surfaces have a coefficient of friction that can be substantially larger than 1. While it

6240-675: Is more commonly performed by an instrument. If one defines the energy per pulse as ε p u l s e = ∫ 0 T p ( t ) d t {\displaystyle \varepsilon _{\mathrm {pulse} }=\int _{0}^{T}p(t)\,dt} then the average power is P a v g = 1 T ∫ 0 T p ( t ) d t = ε p u l s e T . {\displaystyle P_{\mathrm {avg} }={\frac {1}{T}}\int _{0}^{T}p(t)\,dt={\frac {\varepsilon _{\mathrm {pulse} }}{T}}.} One may define

6370-562: Is more useful to define static friction in terms of the maximum angle before which one of the items will begin sliding. This is called the angle of friction or friction angle . It is defined as: tan ⁡ θ = μ s {\displaystyle \tan {\theta }=\mu _{\mathrm {s} }} and thus: θ = arctan ⁡ μ s {\displaystyle \theta =\arctan {\mu _{\mathrm {s} }}} where θ {\displaystyle \theta }

6500-501: Is much heavier than an average of a little over one ton per passenger for an automobile. This means that for an Amtrak passenger train in 1975, much of the energy savings of the lower rolling resistance was lost to its greater weight. An example of a very light high-speed passenger train is the N700 Series Shinkansen , which weighs 715 tonnes and carries 1323 passengers, resulting in a per-passenger weight of about half

6630-479: Is no sliding occurring, the friction force can have any value from zero up to F max {\displaystyle F_{\text{max}}} . Any force smaller than F max {\displaystyle F_{\text{max}}} attempting to slide one surface over the other is opposed by a frictional force of equal magnitude and opposite direction. Any force larger than F max {\displaystyle F_{\text{max}}} overcomes

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6760-437: Is notable that slip does not occur in driven wheels, which are not subjected to driving torque, under different conditions except braking. Therefore, rolling resistance, namely hysteresis loss, is the main source of energy dissipation in driven wheels or axles, whereas in the drive wheels and axles slip resistance, namely loss due to wheel slip, plays the role as well as rolling resistance. Significance of rolling or slip resistance

6890-404: Is noteworthy that V s / Ω {\displaystyle V_{s}/\Omega } is usually not equal to the radius of the rolling body as a result of wheel slip. The slip between wheel and ground inevitably occurs whenever a driving or braking torque is applied to the wheel. Consequently, the linear speed of the vehicle differs from the wheel's circumferential speed. It

7020-456: Is often stated that the COF is a "material property," it is better categorized as a "system property." Unlike true material properties (such as conductivity, dielectric constant, yield strength), the COF for any two materials depends on system variables like temperature , velocity , atmosphere and also what are now popularly described as aging and deaging times; as well as on geometric properties of

7150-434: Is on a level surface and subjected to an external force P {\displaystyle P} tending to cause it to slide, then the normal force between the object and the surface is just N = m g + P y {\displaystyle N=mg+P_{y}} , where m g {\displaystyle mg} is the block's weight and P y {\displaystyle P_{y}}

7280-429: Is only observed by instruments. Such rapid slip may result in excessive wear or damage. Rolling resistance greatly increases with applied torque. At high torques, which apply a tangential force to the road of about half the weight of the vehicle, the rolling resistance may triple (a 200% increase). This is in part due to a slip of about 5%. The rolling resistance increase with applied torque is not linear, but increases at

7410-423: Is perpendicular to the surfaces. In the simple case of a mass resting on a horizontal surface, the only component of the normal force is the force due to gravity, where N = m g {\displaystyle N=mg\,} . In this case, conditions of equilibrium tell us that the magnitude of the friction force is zero , F f = 0 {\displaystyle F_{f}=0} . In fact,

7540-409: Is proportional to the normal force (until saturation, which takes place when all area is in atomic contact); and that the frictional force is proportional to the applied normal force, independently of the contact area. The Coulomb approximation is fundamentally an empirical construct. It is a rule-of-thumb describing the approximate outcome of an extremely complicated physical interaction. The strength of

7670-516: Is reported that the effect of diameter on rolling resistance is negligible (within a practical range of diameters). The driving torque T {\displaystyle T} to overcome rolling resistance R r {\displaystyle R_{r}} and maintain steady speed on level ground (with no air resistance) can be calculated by: T = V s Ω R r {\displaystyle T={\frac {V_{s}}{\Omega }}R_{r}} where It

7800-573: Is responsible for the Coulomb damping of an oscillating or vibrating system. New models are beginning to show how kinetic friction can be greater than static friction. In many other cases roughness effects are dominant, for example in rubber to road friction. Surface roughness and contact area affect kinetic friction for micro- and nano-scale objects where surface area forces dominate inertial forces. The origin of kinetic friction at nanoscale can be rationalized by an energy model. During sliding,

7930-479: Is several times higher than the neglected resistances. The "rolling resistance coefficient" is defined by the following equation:   F = C r r N {\displaystyle \ F=C_{rr}N} where C r r {\displaystyle C_{rr}} is the force needed to push (or tow) a wheeled vehicle forward (at constant speed on a level surface, or zero grade, with zero air resistance) per unit force of weight. It

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8060-466: Is the coefficient of static friction . This is usually larger than its kinetic counterpart. The coefficient of static friction exhibited by a pair of contacting surfaces depends upon the combined effects of material deformation characteristics and surface roughness , both of which have their origins in the chemical bonding between atoms in each of the bulk materials and between the material surfaces and any adsorbed material . The fractality of surfaces,

8190-401: Is the electrical resistance , measured in ohms . In the case of a periodic signal s ( t ) {\displaystyle s(t)} of period T {\displaystyle T} , like a train of identical pulses, the instantaneous power p ( t ) = | s ( t ) | 2 {\textstyle p(t)=|s(t)|^{2}} is also

8320-412: Is the angle from horizontal and μ s is the static coefficient of friction between the objects. This formula can also be used to calculate μ s from empirical measurements of the friction angle. Determining the forces required to move atoms past each other is a challenge in designing nanomachines . In 2008 scientists for the first time were able to move a single atom across a surface, and measure

8450-497: Is the downward component of the external force. Prior to sliding, this friction force is F f = − P x {\displaystyle F_{f}=-P_{x}} , where P x {\displaystyle P_{x}} is the horizontal component of the external force. Thus, F f ≤ μ N {\displaystyle F_{f}\leq \mu N} in general. Sliding commences only after this frictional force reaches

8580-499: Is the limiting value of the average power as the time interval Δ t approaches zero. P = lim Δ t → 0 P a v g = lim Δ t → 0 Δ W Δ t = d W d t . {\displaystyle P=\lim _{\Delta t\to 0}P_{\mathrm {avg} }=\lim _{\Delta t\to 0}{\frac {\Delta W}{\Delta t}}={\frac {dW}{dt}}.} When power P

8710-465: Is the product of the torque τ and angular velocity ω , P ( t ) = τ ⋅ ω , {\displaystyle P(t)={\boldsymbol {\tau }}\cdot {\boldsymbol {\omega }},} where ω is angular frequency , measured in radians per second . The ⋅ {\displaystyle \cdot } represents scalar product . In fluid power systems such as hydraulic actuators, power

8840-555: Is the same as kilograms (kg force) per metric ton (tonne = 1000 kg ), which is the same as pounds of resistance per 1000 pounds of load or Newtons/kilo-Newton, etc. For the US railroads, lb/ton has been traditionally used; this is just 2000 C r r {\displaystyle 2000C_{rr}} . Thus, they are all just measures of resistance per unit vehicle weight. While they are all "specific resistances", sometimes they are just called "resistance" although they are really

8970-471: Is transformed to heat as well as the free energy of the structural changes and other types of dissipation , so mechanical energy is not conserved. The complexity of the interactions involved makes the calculation of friction from first principles difficult and it is often easier to use empirical methods for analysis and the development of theory. There are several types of friction: Many ancient authors including Aristotle , Vitruvius , and Pliny

9100-537: Is usually higher than the coefficient of kinetic friction. Static friction is considered to arise as the result of surface roughness features across multiple length scales at solid surfaces. These features, known as asperities are present down to nano-scale dimensions and result in true solid to solid contact existing only at a limited number of points accounting for only a fraction of the apparent or nominal contact area. The linearity between applied load and true contact area, arising from asperity deformation, gives rise to

9230-860: The atomic scale , showing that, on that scale, dry friction is the product of the inter-surface shear stress and the contact area. These two discoveries explain Amonton's first law (below) ; the macroscopic proportionality between normal force and static frictional force between dry surfaces. The elementary property of sliding (kinetic) friction were discovered by experiment in the 15th to 18th centuries and were expressed as three empirical laws: Dry friction resists relative lateral motion of two solid surfaces in contact. The two regimes of dry friction are 'static friction' (" stiction ") between non-moving surfaces, and kinetic friction (sometimes called sliding friction or dynamic friction) between moving surfaces. Coulomb friction, named after Charles-Augustin de Coulomb ,

9360-400: The coefficient of kinetic friction . The Coulomb friction is equal to F f {\displaystyle F_{\mathrm {f} }} , and the frictional force on each surface is exerted in the direction opposite to its motion relative to the other surface. Arthur Morin introduced the term and demonstrated the utility of the coefficient of friction. The coefficient of friction

9490-414: The fundamental theorem of calculus , we know that P = d W d t = d d t ∫ Δ t F ⋅ v d t = F ⋅ v . {\displaystyle P={\frac {dW}{dt}}={\frac {d}{dt}}\int _{\Delta t}\mathbf {F} \cdot \mathbf {v} \,dt=\mathbf {F} \cdot \mathbf {v} .} Hence

9620-424: The mechanical advantage of the system (output force per input force) is given by M A = F B F A = v A v B . {\displaystyle \mathrm {MA} ={\frac {F_{\text{B}}}{F_{\text{A}}}}={\frac {v_{\text{A}}}{v_{\text{B}}}}.} The similar relationship is obtained for rotating systems, where T A and ω A are

9750-565: The movie Archived 2015-01-10 at the Wayback Machine for more details. Power (physics)#Mechanical power Power is the amount of energy transferred or converted per unit time. In the International System of Units , the unit of power is the watt , equal to one joule per second. Power is a scalar quantity. Specifying power in particular systems may require attention to other quantities; for example,

9880-429: The viscoelastic characteristics of the rubber. This main principle is illustrated in the figure of the rolling cylinders. If two equal cylinders are pressed together then the contact surface is flat. In the absence of surface friction, contact stresses are normal (i.e. perpendicular) to the contact surface. Consider a particle that enters the contact area at the right side, travels through the contact patch and leaves at

10010-404: The 'song' of a glass harp , phenomena which involve stick and slip, modelled as a drop of friction coefficient with velocity. A practically important case is the self-oscillation of the strings of bowed instruments such as the violin , cello , hurdy-gurdy , erhu , etc. A connection between dry friction and flutter instability in a simple mechanical system has been discovered, watch

10140-435: The 20th century has been to understand the physical mechanisms behind friction. Frank Philip Bowden and David Tabor (1950) showed that, at a microscopic level , the actual area of contact between surfaces is a very small fraction of the apparent area. This actual area of contact, caused by asperities increases with pressure. The development of the atomic force microscope (ca. 1986) enabled scientists to study friction at

10270-664: The Elder , were interested in the cause and mitigation of friction. They were aware of differences between static and kinetic friction with Themistius stating in 350 A.D. that "it is easier to further the motion of a moving body than to move a body at rest". The classic laws of sliding friction were discovered by Leonardo da Vinci in 1493, a pioneer in tribology , but the laws documented in his notebooks were not published and remained unknown. These laws were rediscovered by Guillaume Amontons in 1699 and became known as Amonton's three laws of dry friction. Amontons presented

10400-440: The above definition of C r r {\displaystyle C_{rr}} that the rolling resistance is directly proportional to vehicle weight but it is not . There are at least two popular models for calculating rolling resistance. The results of these tests can be hard for the general public to obtain as manufacturers prefer to publicize "comfort" and "performance". The coefficient of rolling resistance for

10530-406: The approximation is its simplicity and versatility. Though the relationship between normal force and frictional force is not exactly linear (and so the frictional force is not entirely independent of the contact area of the surfaces), the Coulomb approximation is an adequate representation of friction for the analysis of many physical systems. When the surfaces are conjoined, Coulomb friction becomes

10660-421: The average power P avg over that period is given by the formula P a v g = Δ W Δ t . {\displaystyle P_{\mathrm {avg} }={\frac {\Delta W}{\Delta t}}.} It is the average amount of work done or energy converted per unit of time. Average power is often called "power" when the context makes it clear. Instantaneous power

10790-583: The beginning and end of the path along which the work was done. The power at any point along the curve C is the time derivative: P ( t ) = d W d t = F ⋅ v = − d U d t . {\displaystyle P(t)={\frac {dW}{dt}}=\mathbf {F} \cdot \mathbf {v} =-{\frac {dU}{dt}}.} In one dimension, this can be simplified to: P ( t ) = F ⋅ v . {\displaystyle P(t)=F\cdot v.} In rotational systems, power

10920-435: The classic empirical model of friction (static, kinetic, and fluid) commonly used today in engineering. In 1877, Fleeming Jenkin and J. A. Ewing investigated the continuity between static and kinetic friction. In 1907, G.H. Bryan published an investigation of the foundations of thermodynamics, Thermodynamics: an Introductory Treatise dealing mainly with First Principles and their Direct Applications . He noted that for

11050-501: The contacting materials, such as surface roughness. The coefficient of friction is not a function of mass or volume. For instance, a large aluminum block has the same coefficient of friction as a small aluminum block. However, the magnitude of the friction force itself depends on the normal force, and hence on the mass of the block. Depending on the situation, the calculation of the normal force N {\displaystyle N} might include forces other than gravity. If an object

11180-433: The direction of the frictional force against a surface is opposite to the motion that surface would experience in the absence of friction. Thus, in the static case, the frictional force is exactly what it must be in order to prevent motion between the surfaces; it balances the net force tending to cause such motion. In this case, rather than providing an estimate of the actual frictional force, the Coulomb approximation provides

11310-433: The drive wheels of an accelerating car experience a frictional force pointing forward; if they did not, the wheels would spin, and the rubber would slide backwards along the pavement. Note that it is not the direction of movement of the vehicle they oppose, it is the direction of (potential) sliding between tire and road. The normal force is defined as the net force compressing two parallel surfaces together, and its direction

11440-403: The driving wheel(s) becomes greater than the vehicle speed due to slippage. Since power is equal to force times velocity and the wheel velocity has increased, the power required has increased accordingly. The pure "rolling resistance" for a train is that which happens due to deformation and possible minor sliding at the wheel-road contact. For a rubber tire, an analogous energy loss happens over

11570-406: The element and of the voltage across the element. Power is the rate with respect to time at which work is done; it is the time derivative of work : P = d W d t , {\displaystyle P={\frac {dW}{dt}},} where P is power, W is work, and t is time. We will now show that the mechanical power generated by a force F on a body moving at

11700-410: The energy dissipated by vibration and oscillation of both the roadbed and the vehicle, and sliding of the wheel on the roadbed surface (pavement or a rail). But there is an even broader sense that would include energy wasted by wheel slippage due to the torque applied from the engine . This includes the increased power required due to the increased velocity of the wheels where the tangential velocity of

11830-418: The energy to overcome this broad-sense rolling resistance. For tires, rolling resistance is defined as the energy consumed by a tire per unit distance covered. It is also called rolling friction or rolling drag. It is one of the forces that act to oppose the motion of a driver. The main reason for this is that when the tires are in motion and touch the surface, the surface changes shape and causes deformation of

11960-401: The entire tire, but it is still called "rolling resistance". In the broad sense, "rolling resistance" includes wheel bearing resistance, energy loss by shaking both the roadbed (and the earth underneath) and the vehicle itself, and by sliding of the wheel, road/rail contact. Railroad textbooks seem to cover all these resistance forces but do not call their sum "rolling resistance" (broad sense) as

12090-459: The extent of the surface area; the normal pressure (or load); and the length of time that the surfaces remained in contact (time of repose). Coulomb further considered the influence of sliding velocity, temperature and humidity, in order to decide between the different explanations on the nature of friction that had been proposed. The distinction between static and dynamic friction is made in Coulomb's friction law (see below), although this distinction

12220-611: The force is variable over a three-dimensional curve C , then the work is expressed in terms of the line integral: W = ∫ C F ⋅ d r = ∫ Δ t F ⋅ d r d t   d t = ∫ Δ t F ⋅ v d t . {\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {r} =\int _{\Delta t}\mathbf {F} \cdot {\frac {d\mathbf {r} }{dt}}\ dt=\int _{\Delta t}\mathbf {F} \cdot \mathbf {v} \,dt.} From

12350-556: The force of gravity is perpendicular to the face of the plane. The normal force and the frictional force are ultimately determined using vector analysis, usually via a free body diagram . In general, process for solving any statics problem with friction is to treat contacting surfaces tentatively as immovable so that the corresponding tangential reaction force between them can be calculated. If this frictional reaction force satisfies F f ≤ μ N {\displaystyle F_{f}\leq \mu N} , then

12480-406: The force of static friction and causes sliding to occur. The instant sliding occurs, static friction is no longer applicable—the friction between the two surfaces is then called kinetic friction. However, an apparent static friction can be observed even in the case when the true static friction is zero. An example of static friction is the force that prevents a car wheel from slipping as it rolls on

12610-739: The force per the rolling resistance coefficient , and solving for b {\displaystyle b} , gives b {\displaystyle b} = C r r r {\displaystyle C_{rr}r} . Therefore, if a source gives rolling resistance coefficient ( C r r {\displaystyle C_{rr}} ) as a dimensionless coefficient, it can be converted to b {\displaystyle b} , having units of length, by multiplying C r r {\displaystyle C_{rr}} by wheel radius r {\displaystyle r} . Table of rolling resistance coefficient examples: [3] For example, in earth gravity,

12740-402: The forces required. Using ultrahigh vacuum and nearly zero temperature (5 K), a modified atomic force microscope was used to drag a cobalt atom, and a carbon monoxide molecule, across surfaces of copper and platinum . The Coulomb approximation follows from the assumptions that: surfaces are in atomically close contact only over a small fraction of their overall area; that this contact area

12870-523: The formula is valid for any general situation. In older works, power is sometimes called activity . The dimension of power is energy divided by time. In the International System of Units (SI), the unit of power is the watt (W), which is equal to one joule per second. Other common and traditional measures are horsepower (hp), comparing to the power of a horse; one mechanical horsepower equals about 745.7 watts. Other units of power include ergs per second (erg/s), foot-pounds per minute, dBm ,

13000-489: The friction force always satisfies F f ≤ μ N {\displaystyle F_{f}\leq \mu N} , with equality reached only at a critical ramp angle (given by tan − 1 ⁡ μ {\displaystyle \tan ^{-1}\mu } ) that is steep enough to initiate sliding. The friction coefficient is an empirical (experimentally measured) structural property that depends only on various aspects of

13130-486: The frictional heating is removed rapidly, the temperature drops, the pin remains solid and the COF rises to that of a 'low speed' test. In systems with significant non-uniform stress fields, because local slip occurs before the system slides, the macroscopic coefficient of static friction depends on the applied load, system size, or shape; Amontons' law is not satisfied macroscopically. Under certain conditions some materials have very low friction coefficients. An example

13260-441: The ground. Even though the wheel is in motion, the patch of the tire in contact with the ground is stationary relative to the ground, so it is static rather than kinetic friction. Upon slipping, the wheel friction changes to kinetic friction. An anti-lock braking system operates on the principle of allowing a locked wheel to resume rotating so that the car maintains static friction. The maximum value of static friction, when motion

13390-457: The inclined plane of successive asperities , then why is it not balanced through descending the opposite slope? Leslie was equally skeptical about the role of adhesion proposed by Desaguliers, which should on the whole have the same tendency to accelerate as to retard the motion. In Leslie's view, friction should be seen as a time-dependent process of flattening, pressing down asperities, which creates new obstacles in what were cavities before. In

13520-411: The interface between the materials, namely surface structure . For example, a copper pin sliding against a thick copper plate can have a COF that varies from 0.6 at low speeds (metal sliding against metal) to below 0.2 at high speeds when the copper surface begins to melt due to frictional heating. The latter speed, of course, does not determine the COF uniquely; if the pin diameter is increased so that

13650-424: The left side. Initially its vertical deformation is increasing, which is resisted by the hysteresis effect. Therefore, an additional pressure is generated to avoid interpenetration of the two surfaces. Later its vertical deformation is decreasing. This is again resisted by the hysteresis effect. In this case this decreases the pressure that is needed to keep the two bodies separate. The resulting pressure distribution

13780-530: The linearity between static frictional force and normal force, found for typical Amonton–Coulomb type friction. The static friction force must be overcome by an applied force before an object can move. The maximum possible friction force between two surfaces before sliding begins is the product of the coefficient of static friction and the normal force: F max = μ s F n {\displaystyle F_{\text{max}}=\mu _{\mathrm {s} }F_{\text{n}}} . When there

13910-498: The long course of the development of the law of conservation of energy and of the first law of thermodynamics , friction was recognised as a mode of conversion of mechanical work into heat . In 1798, Benjamin Thompson reported on cannon boring experiments. Arthur Jules Morin (1833) developed the concept of sliding versus rolling friction. In 1842, Julius Robert Mayer frictionally generated heat in paper pulp and measured

14040-423: The material properties of the wheel or tire and the surface. For example, a rubber tire will have higher rolling resistance on a paved road than a steel railroad wheel on a steel rail. Also, sand on the ground will give more rolling resistance than concrete . Sole rolling resistance factor is not dependent on speed. The primary cause of pneumatic tire rolling resistance is hysteresis : A characteristic of

14170-731: The maximum performance of a device in terms of velocity ratios determined by its physical dimensions. See for example gear ratios . The instantaneous electrical power P delivered to a component is given by P ( t ) = I ( t ) ⋅ V ( t ) , {\displaystyle P(t)=I(t)\cdot V(t),} where If the component is a resistor with time-invariant voltage to current ratio, then: P = I ⋅ V = I 2 ⋅ R = V 2 R , {\displaystyle P=I\cdot V=I^{2}\cdot R={\frac {V^{2}}{R}},} where R = V I {\displaystyle R={\frac {V}{I}}}

14300-409: The nature of friction in terms of surface irregularities and the force required to raise the weight pressing the surfaces together. This view was further elaborated by Bernard Forest de Bélidor and Leonhard Euler (1750), who derived the angle of repose of a weight on an inclined plane and first distinguished between static and kinetic friction. John Theophilus Desaguliers (1734) first recognized

14430-414: The path C and v is the velocity along this path. If the force F is derivable from a potential ( conservative ), then applying the gradient theorem (and remembering that force is the negative of the gradient of the potential energy) yields: W C = U ( A ) − U ( B ) , {\displaystyle W_{C}=U(A)-U(B),} where A and B are

14560-414: The power involved in moving a ground vehicle is the product of the aerodynamic drag plus traction force on the wheels, and the velocity of the vehicle. The output power of a motor is the product of the torque that the motor generates and the angular velocity of its output shaft. Likewise, the power dissipated in an electrical element of a circuit is the product of the current flowing through

14690-580: The product of a torque on a shaft and the shaft's angular velocity. Mechanical power is also described as the time derivative of work. In mechanics , the work done by a force F on an object that travels along a curve C is given by the line integral : W C = ∫ C F ⋅ v d t = ∫ C F ⋅ d x , {\displaystyle W_{C}=\int _{C}\mathbf {F} \cdot \mathbf {v} \,dt=\int _{C}\mathbf {F} \cdot d\mathbf {x} ,} where x defines

14820-488: The pulse length τ {\displaystyle \tau } such that P 0 τ = ε p u l s e {\displaystyle P_{0}\tau =\varepsilon _{\mathrm {pulse} }} so that the ratios P a v g P 0 = τ T {\displaystyle {\frac {P_{\mathrm {avg} }}{P_{0}}}={\frac {\tau }{T}}} are equal. These ratios are called

14950-479: The roadbed structure. In the broad sense, specific "rolling resistance" (for vehicles) is the force per unit vehicle weight required to move the vehicle on level ground at a constant slow speed where aerodynamic drag (air resistance) is insignificant and also where there are no traction (motor) forces or brakes applied. In other words, the vehicle would be coasting if it were not for the force to maintain constant speed. This broad sense includes wheel bearing resistance,

15080-408: The role of adhesion in friction. Microscopic forces cause surfaces to stick together; he proposed that friction was the force necessary to tear the adhering surfaces apart. The understanding of friction was further developed by Charles-Augustin de Coulomb (1785). Coulomb investigated the influence of four main factors on friction: the nature of the materials in contact and their surface coatings;

15210-405: The same materials. However, Richard Feynman comments that "with dry metals it is very hard to show any difference." The friction force between two surfaces after sliding begins is the product of the coefficient of kinetic friction and the normal force: F k = μ k F n {\displaystyle F_{k}=\mu _{\mathrm {k} }F_{n}} . This

15340-865: The so-called Painlevé paradoxes , may be encountered with Coulomb friction. Dry friction can induce several types of instabilities in mechanical systems which display a stable behaviour in the absence of friction. These instabilities may be caused by the decrease of the friction force with an increasing velocity of sliding, by material expansion due to heat generation during friction (the thermo-elastic instabilities), or by pure dynamic effects of sliding of two elastic materials (the Adams–Martins instabilities). The latter were originally discovered in 1995 by George G. Adams and João Arménio Correia Martins for smooth surfaces and were later found in periodic rough surfaces. In particular, friction-related dynamical instabilities are thought to be responsible for brake squeal and

15470-462: The square root of the load. If Crr is itself dependent on wheel load per an inverse square-root rule, then for an increase in load of 2% only a 1% increase in rolling resistance occurs. For pneumatic tires, the direction of change in Crr (rolling resistance coefficient) depends on whether or not tire inflation is increased with increasing load. It is reported that, if inflation pressure is increased with load according to an (undefined) "schedule", then

15600-426: The sum of components): Wheel bearing torque losses can be measured as a rolling resistance at the wheel rim, Crr . Railroads normally use roller bearings which are either cylindrical (Russia) or tapered (United States). The specific rolling resistance in bearings varies with both wheel loading and speed. Wheel bearing rolling resistance is lowest with high axle loads and intermediate speeds of 60–80 km/h with

15730-435: The temperature rise. In 1845, Joule published a paper entitled The Mechanical Equivalent of Heat , in which he specified a numerical value for the amount of mechanical work required to "produce a unit of heat", based on the friction of an electric current passing through a resistor, and on the friction of a paddle wheel rotating in a vat of water. Osborne Reynolds (1866) derived the equation of viscous flow. This completed

15860-456: The tentative assumption was correct, and it is the actual frictional force. Otherwise, the friction force must be set equal to F f = μ N {\displaystyle F_{f}=\mu N} , and then the resulting force imbalance would then determine the acceleration associated with slipping. The coefficient of friction (COF), often symbolized by the Greek letter μ ,

15990-409: The tire. For highway motor vehicles, there is some energy dissipated in shaking the roadway (and the earth beneath it), the shaking of the vehicle itself, and the sliding of the tires. But, other than the additional power required due to torque and wheel bearing friction, non-pure rolling resistance doesn't seem to have been investigated, possibly because the "pure" rolling resistance of a rubber tire

16120-744: The torque and angular velocity of the input and T B and ω B are the torque and angular velocity of the output. If there are no losses in the system, then P = T A ω A = T B ω B , {\displaystyle P=T_{\text{A}}\omega _{\text{A}}=T_{\text{B}}\omega _{\text{B}},} which yields the mechanical advantage M A = T B T A = ω A ω B . {\displaystyle \mathrm {MA} ={\frac {T_{\text{B}}}{T_{\text{A}}}}={\frac {\omega _{\text{A}}}{\omega _{\text{B}}}}.} These relations are important because they define

16250-428: The total energy expenditure of the world. As briefly discussed later, there are many different contributors to the retarding force in friction, ranging from asperity deformation to the generation of charges and changes in local structure . Friction is not itself a fundamental force , it is a non-conservative force – work done against friction is path dependent. In the presence of friction, some mechanical energy

16380-436: The tractive effort is so high that the wheel is close to substantial slipping (more than just a few percent as discussed above), then slip rapidly increases with tractive effort and is no longer linear. With a little higher applied tractive effort the wheel spins out of control and the adhesion drops resulting in the wheel spinning even faster. This is the type of slipping that is observable by eye—the slip of say 2% for traction

16510-417: The two coefficients are equal, such as teflon-on-teflon. Most dry materials in combination have friction coefficient values between 0.3 and 0.6. Values outside this range are rarer, but teflon , for example, can have a coefficient as low as 0.04. A value of zero would mean no friction at all, an elusive property. Rubber in contact with other surfaces can yield friction coefficients from 1 to 2. Occasionally it

16640-402: The value F f = μ N {\displaystyle F_{f}=\mu N} . Until then, friction is whatever it needs to be to provide equilibrium, so it can be treated as simply a reaction. If the object is on a tilted surface such as an inclined plane, the normal force from gravity is smaller than m g {\displaystyle mg} , because less of

16770-909: The velocity v can be expressed as the product: P = d W d t = F ⋅ v {\displaystyle P={\frac {dW}{dt}}=\mathbf {F} \cdot \mathbf {v} } If a constant force F is applied throughout a distance x , the work done is defined as W = F ⋅ x {\displaystyle W=\mathbf {F} \cdot \mathbf {x} } . In this case, power can be written as: P = d W d t = d d t ( F ⋅ x ) = F ⋅ d x d t = F ⋅ v . {\displaystyle P={\frac {dW}{dt}}={\frac {d}{dt}}\left(\mathbf {F} \cdot \mathbf {x} \right)=\mathbf {F} \cdot {\frac {d\mathbf {x} }{dt}}=\mathbf {F} \cdot \mathbf {v} .} If instead

16900-412: Was already drawn by Johann Andreas von Segner in 1758. The effect of the time of repose was explained by Pieter van Musschenbroek (1762) by considering the surfaces of fibrous materials, with fibers meshing together, which takes a finite time in which the friction increases. John Leslie (1766–1832) noted a weakness in the views of Amontons and Coulomb: If friction arises from a weight being drawn up

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