Transcript Friction
Friction Friction Problem Situations Physics Montwood High School R. Casao Friction • Friction Ff is a force that resists motion • Friction involves objects in contact with each other. • Friction must be overcome before motion occurs. • Friction is caused by the uneven surfaces of the touching objects. As surfaces are pressed together, they tend to interlock and offer resistance to being moved over each other. Microscopic Friction Surface Roughness Magnified section of a polished steel surface showing surface bumps about 5 x 10-7 m (500 nm) high, which corresponds to several thousand atomic diameters. Adhesion Computer graphic from a simulation showing gold atoms (below) adhering to the point of a sharp nickel probe (above) that has been in contact with the gold surface. Friction • Frictional forces are always in the direction that is opposite to the direction of motion or to the net force that produces the motion. • Friction acts parallel to the surfaces in contact. Types of Friction • Static friction: maximum frictional force between stationary objects. • Until some maximum value is reached and motion occurs, the frictional force is whatever force is necessary to prevent motion. • Static friction will oppose a force until such time as the object “breaks away” from the surface with which it is in contact. • The force that is opposed is that component of an applied force that is parallel to the surface of contact. Types of Friction • The magnitude of the static friction force Ffs has a maximum value which is given by: Ff s s FN • where μs is the coefficient of static friction and FN is the magnitude of the normal force on the body from the surface. Types of Friction • Sliding or kinetic friction: frictional force between objects that are sliding with respect to one another. • Once enough force has been applied to the object to overcome static friction and get the object to move, the friction changes to sliding (or kinetic) friction. • Sliding (kinetic) friction is less than static friction. • If the component of the applied force on the object (parallel to the surface) exceeds Ffs then the magnitude of the opposing force decreases rapidly to a value Fk given by: Fk k FN where μk is the coefficient of kinetic friction. Static Friction The static frictional force keeps an object from starting to move when a force is applied. The static frictional force has a maximum value, but may take on any value from zero to the maximum, depending on what is needed to keep the sum of forces zero. Types of Friction • From 0 to the maximum value of the static frictional force Fs in the figure, the applied force is resisted by the static frictional force until “breakaway”. • Then the sliding (kinetic) frictional force Fk is approximately constant. Types of Friction • Static and sliding friction are dependent on: • The nature of the surfaces in contact. Rough surfaces tend to produce more friction. • The normal force (Fn) pressing the surfaces together; the greater Fn is, the more friction there is. Friction vs. Area Question: Why doesn’t friction depend on contact area? The microscopic area of contact between a box and the floor is only a small fraction of the macroscopic area of the box’s bottom surface. If the box is turned on its side, the macroscopic area is increased, but the microscopic area of contact remains the same (because the contact is more distributed). Therefore the frictional force f is independent of contact area. Types of Friction • Rolling friction: involves one object rolling over a surface or another object. • Fluid friction: involves the movement of a fluid over an object (air resistance or drag in water) or the addition of a lubricant (oil, grease, etc.) to change sliding or rolling friction to fluid friction. Coefficient of Friction • Coefficient of friction (): ratio of the frictional force to the normal force pressing the surfaces together. has no units. • Static: F μs fs Fn • Sliding (kinetic): μ Ffk k Fn •The maximum frictional force is 50 N. As the applied force increases from 0 N to 50 N, the frictional force also increases from 0 N to 50 N and will be equal to the applied force as it increases. •Once the static frictional force of 50 N has been overcome, only a 40 N force is needed to overcome the 40 N kinetic frictional force and produce constant velocity (a = 0 m/s2). •As the applied force increases beyond 40 N, the kinetic frictional force remains at 40 N and the 100 N block will accelerate. A Model of Friction Friction Static Friction Kinetic Friction Fpush f k k FN Kinetic Friction and Speed The kinetic frictional force is also independent of the relative speed of the surfaces, and of their area of contact. Rolling Friction Horizontal Surface – Constant Speed •Constant speed: a = O m/s2. •The normal force pressing the surfaces together is the weight; Fn = Fw ΣFx m a Fx Ff m a Ff Ff μk Fn Fw Fx Ff 0 N Ff μ k Fw Fx Ff Fx Ff μ k Fw Horizontal Surface: a > O m/s2 Fx Ff ΣFx m a Fx Ff m a Fn Fw Ff Ff μk Fn Fw Ff μ k Fw Horizontal Surface: a > O m/s2 • If solving for: • Fx: Fx m a Ff Fx m a μ k Fw Fx m a μ k m g • F f: Ff Fx m a • a: Fx Ff a m Horizontal Surface: Skidding to a Stop or Slowing Down (a < O m/s2) • The frictional force is responsible for the negative acceleration. • Generally, there is no Fx. Ff m a Fn Fw Ff Ff μk Fn Fw Ff μ k Fw Horizontal Surface: Skidding to a Stop or Slowing Down (a < O m/s2) • Most common use involves finding acceleration with a velocity equation and finding k: 2 2 v f v i (2 a Δx ) 2 Δx (v i t ) (0.5 a t ) v f v i (a t ) • Acceleration will be negative because the speed is decreasing. Horizontal Surface: Skidding to a Stop or Slowing Down (a < O m/s2) Ff Ff m a a μk Fn Fw mg g • The negative sign for acceleration a is dropped because k is a ratio of forces that does not depend on direction. • Maximum stopping distance occurs when the tire is rotating. When this happens, a = -s·g. • Otherwise, use a = -k·g to find the acceleration, then use a velocity equation to find distance, time, or speed. Friction, Cars, & Antilock Brakes The diagram shows forces acting on a car with front-wheel drive. Typically, Fn > Fn’ because the engine is over the front wheels. The largest frictional force fs the tire can exert on the road is µs·Fn. Attempts to make the tire exert a force larger than this causes the tire to “burn rubber” and actually reduces the force, since µk<µs. Note that while all points on the rolling tire have the same speed v in the reference frame of the car, in the reference frame of the road the bottom of the tire is at rest, while top is moving forward with a speed of 2·v. Antilock brakes sense the wheel rotation and “ease off” if it close to stopping, maintaining static friction with the road and allowing better control of steering than if the wheels were locked. Antilock Brakes Example: The Effect of Antilock Brakes A car is traveling at 30 m/s along a horizontal road. The coefficients of friction are ms=0.50 and mk=0.40. (a) What is the braking distance xa with antilock brakes? (b) What is the braking distance xb if the brakes lock? 2 v v 2 v02 2 a x and v 0, so x 0 2a aa s g and ab k g v02 (30 m/s)2 xa 91.7 m 2 2 s g 2 (0.5) (9.81 m/s ) v02 (30 m/s)2 xb 114.7 m 2 2 k g 2 (0.4) (9.81 m/s ) Example: A Game of Shuffleboard A cruise-ship passenger uses a shuffleboard cue to push a shuffleboard disk of mass 0.40 kg horizontally along the deck, so that the disk leaves the cue at a speed of 8.5 m/s. The disk then slides a distance of 8.0 m. f k k Fn What is the coefficient of kinetic friction between the disk and deck? F y m ay 0 Fn m g 0 Fn m g F x m ax f k k m g m ax a x k g vx2 v02x 2 ax x 0 v02x 2 k g x v02x (8.5 m/s)2 k 0.46 2 2 g x 2 (9.81 m/s ) (8.0 m) Down an Inclined Plane Down an Inclined Plane • Resolve Fw into Fx and Fy. • The angle of the incline is always equal to the angle between Fw and Fy. • Fw is always the hypotenuse of the right triangle formed by Fw, Fx, and Fy. cos θ Fy Fw Fx sin θ Fw Fy Fw cos θ Fx Fw sin θ Down an Inclined Plane • The force pressing the surfaces together is NOT Fw, but Fy; Fn = Fy. ΣF m a or Fx Ff m a Fx Ff a m Ff Ff μk Fn Fy Ff μ k Fy m g sin m g cos m a mass m cancels out (g sin ) ( g cos ) a (g sin ) a g cos Down an Inclined Plane • If we place an object on an inclined plane and increase the tilt angle to the point at which the object just begins to slide. • What is the relation between and the static coefficient of friction µs? y - axis : FN Fw cos x - axis : f s s FN Fw sin fs Fw sin s Fn Fw cos sin s tan cos Down an Inclined Plane • If the object slides down the incline at constant speed (a = 0 m/s2), the relation between and the kinetic coefficient of friction µk: Fx Ff m 0 m s 2 Fx Ff 0 N Fx Ff Ff Fx Fw sin θ sin θ μk tan θ Fn Fy Fw cos θ cos θ μ k tan θ Down an Inclined Plane • To determine the angle of the incline: • If moving: 1 μk 1 μs θ tan • If at rest: θ tan Example: A Sliding Coin A hardcover book is resting on a tabletop with its front cover facing upward. You place a coin on the cover and very slowly open the book until the coin starts to slide. The angle is the angle of the cover just before the coin begins to slide. Find the coefficient of static friction µs between the coin and book. F y m ay Fn m g cos 0 or FN m g cos f s s FN at , so f s s m g cos F x m ax m g sin f s 0 or f s m g sin Therefore, s cos sin or s tan Example: Dumping a file cabinet Steel on dry steel Free-body diagram A 50.0 kg steel file cabinet is in the back of a dump truck. The truck’s bed, also made of steel, is slowly tilted. What is the size of the static friction force when the truck’s bed is tilted by 20°? At what angle will the file cabinet begin to slide? Example: Dumping a file cabinet F F x w sin f s m g sin f s 0; y n w cos n m g cos 0; f s m g sin (50.0 kg) (9.80 m/s 2 ) sin 20 168 N ; File cabinet will begin to slide when: f s f s max s n s m g cos ; m g sin f s m g sin s m g cos 0; sin s tan ; arctan s arctan(0.80) 38.7 cos Non-Parallel Applied Force on Ramp If an applied force acts on the box at an angle above the horizontal, resolve FA into parallel and perpendicular components FA using the angle + : FA ·sin( + ) N FA ·cos ( + θ) and FA ·sin ( + θ) FA ·cos( + ) fk m·g ·sin FA serves to increase acceleration directly and indirectly: directly by FA ·cos ( + θ) pulling the box down the ramp, and indirectly by FA ·sin ( + θ) lightening the normal support force with the ramp (thereby reducing friction). m·g m·g ·cos Non-Parallel Applied Force on Ramp FA ·sin ( + ) FA N FA ·cos( + ) fk If FA ·sin( + ) is not big enough to lift the box off the ramp, there is no acceleration in the perpendicular direction. So, FA ·sin( + ) + FN = m·g·cos. Remember, FN is what a scale would read if placed under the box, and a scale reads less if a force lifts up on the box. So, FN = m·g ·cos - FA ·sin( + ), which means fk = k ·FN = k ·[m·g ·cos - FA ·sin( + )]. m·g ·sin m·g Non-Parallel Applied Force on Ramp FA ·sin( + ) FA N FA ·cos( + ) fk m·g ·sin If the combined force of FA ·cos( + ) + m·g ·sin is is enough to move the box: FA ·cos( + ) + m·g·sin - k ·[m·g·cos - FA ·sin( + )] = m·a m·g ·cos m·g Up an Inclined Plane Up an Inclined Plane • Resolve Fw into Fx and Fy. • The angle of the incline is always equal to the angle between Fw and Fy. • Fw is always the hypotenuse of the right triangle formed by Fw, Fx, and Fy. Fy F cos θ Fw sin θ Fy Fw cos θ Fx Fw sin θ x Fw Up an Inclined Plane • Fa is the force that must be applied in the direction of motion. • Fa must overcome both friction and the x-component of the weight. • The force pressing the surfaces together is Fy. Up an Inclined Plane Fn Fy ΣFx m a Fa Ff Fx m a Fa Ff Fx a m Ff Ff μk Fn Fy Ff μ k Fy •For constant speed, a = 0 m/s2. Fa = Fx + Ff •For a > 0 m/s2. Fa = Fx + Ff + (m·a) Pulling an Object on a Flat Surface Pulling an Object on a Flat Surface •The pulling force F is resolved into Fx and Fy. Fx cos θ F Fy sin θ F Fx F cos θ Fy F sin θ Pulling an Object on a Flat Surface •Fn is the force that the ground exerts upward on the mass. Fn equals the downward weight Fw minus the upward force Fy from the pulling force. •For constant speed, a = 0 m/s2. ΣFy 0 N Fn Fy Fw 0 N Fn Fw Fy Ff Ff μk Fn Fw Fy Ff μ k (Fw Fy ) ΣFx m a Fx Ff m a Fx Ff a m Example: Pulling A Sled Two children sitting on a sled at rest in the snow ask you to pull them. You pull on the sled’s rope, which makes an angle of 40° with the horizontal. The children have a combined mass of 45 kg, and the sled has a mass of 5.0 kg. The coefficients of static and kinetic friction are µs=0.20 and µk=0.15, and the sled is initially at rest. Find the acceleration of the sled and children if the rope tension is 100 N. F y m ay FN T sin m g m 0 or FN m g T sin FN 50kg 9.8 m 100 N sin 40 425.72 N s2 T cos f k m ax F max k fk f k k FN 0.15 425.72 N 63.86 N FN x 100 N cos 40 63.86 N 50kg ax ax 100 N cos 40 63.86 N 0.2549 m 2 s 50kg Simultaneous Pulling and Pushing an Object on a Flat Surface Simultaneous Pulling and Pushing an Object on a Flat Surface Σ Fy 0 N Fn Fy Fw 0 N Fx cos θ F Fy sin θ F Fx F cos θ Fn Fw Fy Fy F sin θ a μk Ff Ff Fn Fw Fy Ff μ k (Fw Fy ) ΣFx m a Fx Fpush Ff m a Fx Fpush Ff m Pushing an Object on a Flat Surface Pushing an Object on a Flat Surface •The pushing force F is resolved into Fx and Fy. Fx cosθ F Fy sinθ F Fx F cosθ Fy F sinθ Pushing an Object on a Flat Surface •Fn is the force that the ground exerts upward on the mass. Fn equals the downward weight Fw plus the upward force Fy from the pushing force. •For constant speed, a = 0 m/s2. ΣFy 0 N Fn Fy Fw 0 N Fn Fw Fy Ff Ff μk Fn Fw Fy Ff μ k (Fw Fy ) ΣFx m a Fx Ff m a Fx Ff a m Pulling and Tension • The acceleration a of both masses is the same. Pulling and Tension • For each mass: Fn1 Fw1 Fn2 Fw2 Ff1 μ k Fn1 Ff2 μ k Fn2 • Isolate each mass and examine the forces acting on that mass. Pulling and Tension •m1 = mass ΣF m1 a T1 T2 Ff1 m1 a •T1 may not be a tension, but could be an applied force (Fa) that causes motion. Pulling and Tension •m2 = mass ΣF m 2 a T2 Ff 2 m 2 a Pulling and Tension • This problem can often be solved as a system of equations: T1 T2 Ff1 m1 a T2 Ff 2 m 2 a • See the Solving Simultaneous Equations notes for instructions on how to solve this problem using a TI or Casio calculator. Revisiting Tension and Friction Revisiting Tension and Friction •For the hanging mass, •For the mass on the table, m1: m2 : ΣF m2 a ΣF m a Fn1 Fw1 Fw 2 T m2 a Ff μ Fn1 Fw 2 m 2 g m2 g T m2 a •The acceleration a of both masses is the same. T-Ff m1 a Revisiting Tension and Friction m2 g m1 a Ff m2 a m2 g Ff m2 a m1 a m2 g Ff a m2 m1 Example: A Sliding Block A block of mass m2 = 5.0 kg has been adjusted so that the block m1 = 7.0 kg is just on the verge of sliding. (a) What is the coefficient of static friction ms between the table and the block? F FN m1 g m1 ay 0 so FN m1 g y F x T f m1 ax 0 so f s FN s m1 g T F x' m2 g T m2 ax ' 0 so T m2 g m2 g m2 5kg Therefore, s m1 g m2 g ; s 0.71 m2 g m2 7kg Example: A Sliding Block (b) With a slight push, the blocks move with acceleration a. Find a if µk = 0.54. F x T f m1 ax so T k m1 g m1 ax T m1 ax k m1 g F x' m2 g T m2 ax ' T m2 g m2 ax ' ax ax ' a T=T, therefore, m1 a k m1 g m2 g m2 a m1 a m2 a m2 g k m1 g g (m2 k m1 ) a (m1 m2 ) 9.8 m/s 2 5.0 kg 0.54 7.0 kg a 1.0 m/s 2 7.0 kg 5.0 kg Normal Force Not Associated with Weight. • A normal force can exist that is totally unrelated to the weight of an object. friction applied force normal weight FN = applied force Friction is Always Parallel to Surfaces…. •In this case, for the block to remain in position against the wall without moving: • the upward frictional force Ff has to be equal and opposite to the downward weight Fw. •The rightward applied force F has to be equal ad opposite to the leftward normal force FN. Ff F FN FW (0.20)