Frictional Forces. Friction has its basis in surfaces that are not completely smooth: 1/29

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1 Frictional Force Friction ha it bai in urface that are not completely mooth: 1/29

2 Microcopic Friction Surface Roughne Adheion Magnified ection of a polihed teel urface howing urface irregularitie about 5 x 10-7 m (500 nm) high. Thi height correpond to everal thouand atomic diameter Computer graphic from a imulation howing gold atom (below) adhering to the point of a harp nickel probe (above) that ha been in contact with the gold urface. 2/29

3 Friction v. Area Quetion: Why doen t friction depend on contact area? The microcopic area of contact between a box and the floor i only a mall fraction of the macrocopic area of the box bottom urface. If the box i turned on it ide, the macrocopic area i increaed, but the microcopic area of contact remain the ame (becaue the contact i more ditributed). Therefore the frictional force f i independent of contact area. 3/29

4 A Model of Friction Friction 4/29

5 Static Friction 5/29

6 Static Friction The tatic frictional force keep an object from tarting to move when a force i applied. The tatic frictional force ha a maximum value, but may take on any value from zero to the maximum, depending on what i needed to keep the um of force zero. 6/29

7 Static Frictional Force (6-2) where (6-3) The tatic frictional force i alo independent of the area of contact and the relative peed of the urface. 7/29

8 Kinetic Friction F puh f k = µ k n 8/29

9 Kinetic Friction Kinetic friction: the friction experienced by urface liding againt one another The kinetic frictional force depend on the normal force: The contant kinetic friction. i called the coefficient of 9/29

10 Kinetic Friction and Speed The kinetic frictional force i alo independent of the relative peed of the urface, and of their area of contact. 10/29

11 Friction i complicated. A ueful empirical model wa preented by Charle Coulomb in 1781: 1. The force of tatic friction ha a maximum value; if you puh too hard, the block move. Thi maximum value i proportional to the normal force the urface exert on each other. 2. Once the object i liding, kinetic friction i approximately independent of velocity, and uually maller than the maximum tatic friction force. The force of kinetic friction i alo proportional to the normal force. 11

12 Define two pure number (no unit): µ ( coefficient of tatic friction ) µ k ( coefficient of kinetic friction ) ( µ i a Greek letter, pronounced mu ) Then Coulomb rule are: f f k = µ n = µ n k Quetion : would it be correct to write thee a vector equation,? f = µ n F A mg n f 12

13 µ k µ Copper on teel Aluminum on aluminum Teflon on Teflon Value depend on moothne, temperature, etc. and are approximate Uually µ < 1, but not alway Uually, µ k i le than µ, and never larger The coefficient depend on the material, but not on the urface area, contact preure, etc. 13

14 Coefficient of Friction f µ F n f max = µ F n f k = µ k F n Typically, µ >µ k 14/29

15 Rolling Friction 15/29

16 Example: A Game of Shuffleboard A cruie-hip paenger ue a huffleboard cue to puh a huffleboard dik of ma 0.40 kg horizontally along the deck, o that the dik leave the cue at a peed of 8.5 m/. The dik then lide a ditance of 8.0 m. What i the coefficient of kinetic friction between the dik and deck? F y = ma = y F mg = 0 F = mg n 0 n f = µ F k k n Fx = max f = µ mg = ma ax = µ kg k k x v = v + 2a Δx 0= v 2µ gδx x 0x x 0x k 2 2 v0 x (8.5 m/) µ k = = = gΔx 2(9.81 m/ )(8.0 m) 16/29

17 Clicker Quetion 1 You puh two identical brick acro a tabletop with contant peed v. Which Cae, a hown above, require more force F n? a. F 1 >F 2 b. F 1 =F 2 c. F 1 <F 2 d. Need to know brick area to anwer 17/29

18 Inclined Plane Friction Suppoe you place an object on an inclined plane and increae the tilt angle, noting the angle θ at which the object jut begin to lide. What i the relation between θ and the tatic coefficient of friction µ? y-axi: x-axi: n= wcoθ f = nµ = winθ wcoθµ = winθ inθ µ = = tanθ coθ 18/29

19 Example: A Sliding Coin A hardcover book i reting on a tabletop with it front cover facing upward. You place a coin on the cover and very lowly open the book until the coin tart to lide. The angle θ max (known a the angle or repoe) i the angle of the cover jut before the coin begin to lide. Find the coefficient of tatic friction µ between the coin and book. F = ma F mgcoθ = 0 or F = mgcoθ y y n n f = µ F at θ, o f = µ mgcoθ n max F = ma f + mginθ = 0 or f = mginθ max x x max Therefore, µ coθ = in θ or µ = tanθ max max max max 19/29

20 Example: Pulling A Sled Two children itting on a led at ret in the now ak you to pull them. You oblige by pulling on the led rope, which make an angle of 40 with the horizontal. The children have a combined ma of 45 kg, and the led ha a ma of 5.0 kg. The coefficient of tatic and kinetic friction are µ =0.20 and µ k =0.15, and the led i initially at ret. Find the acceleration of the led and children (a) if the rope tenion i 100 N, and (b) if it i 140 N. F = ma F mg+ Tinθ = 0 or F = mg Tinθ y y n n If led doe not move, Fx = max f + Tcoθ = 0 or f = µ Fn = Tcoθ µ mg µ ( mg T in θ) = T co θ; to move led, Tmin = = N, o aa = 0 coθ + µ inθ If led move, F = ma f + Tcoθ = ma x x k b a = [ µ ( mg Tin θ) + Tco θ]/ m= m/ b k 2 20/29

21 Example: A Sliding Block A block of ma m 2 = 5.0 kg ha been adjuted o that the block m 1 = 7.0 kg i jut on the verge of liding. (a) What i the coefficient of tatic friction µ between the table and the block? F = F mg = ma = 0 o F = mg y n 1 1 y n 1 F = m g T = m a = 0 o T = m g x' 2 2 x' 2 (b) With a light nudge, the block move with acceleration a. Find a if µ k = F = f + T = ma = 0 o f = µ F = µ mg = T x 1 x n 1 Therefore, µ = m2/ m1 = (5.0 kg) /(7.0 kg) = 0.71 F = f + T = ma o ma = T µ mg x 1 x 1 x k 1 Therefore, a= g( m µ m ) /( m + m ) 2 k F = m g T = m a ; a = a = a x' 2 2 x' x x' = (9.81 m/ )[(5.0 kg) (0.54)(7.0 kg)]/[(7.0 kg) + (5.0 kg)] = 1.0 m/ /29

22 Example: The Runaway Buggy A runaway baby buggy of ma m B i liding without friction acro a frozen pond toward a hole in the ice. You have ma m Y and race after the buggy on ice kate. A you grab it, you and the buggy are a ditance D from the hole and moving with peed v 0 toward it. The kinetic friction coefficient between your kate and the ice i µ k. (a) What i the lowet value of D that would allow you to top the buggy? F = F m g = a = 0 o F = m g yy ny Y y ny Y F = f + F = m a o µ m g+ F = m a µ kmy FYB + FBY µ kmy g = my ax + mbax ax = g m + m xy IYk BY Y x k Y BY Y x (b) What force do you exert on the buggy a you low it? Y B k Y B FYB = mb ax = g m Y + m B F = F = m a, and F = F xb YB B x BY YB = xδ or = 0 v v a x D µ mm v (1 + mb / my) 2gµ k 22/29

23 Example: Pulling a Toboggan A child of ma m C it on a toboggan of ma m T, which in turn it on a frictionle frozen pond. The toboggan i pulled with horizontal force F ap. The friction coefficient between the child and toboggan are µ and µ k. (a) What i maximum value of F ap uch that the child will not lide off the toboggan? F = F m g = m a = 0 o F = m g F = f = m a Cy TCn C C Cy TCn C Cx TC C Cx F F f m a Tx = ap CT = T Tx fct max = µ mc g, fct = ftc, and acx = atx = ax µ mg= ma o a = µ g F µ m g = m µ g or F = µ g( m + m ) C C Cx x ap C T ap C T (b) Find the acceleration of the child (a C ) and toboggan (a T ) when the force i larger than that value. F = F = F = m g and f = f = f = µ F = µ m g CTn CTn n C CTx TCx k k n k C F = f = m a o f = µ F = µ m g = m a and a = µ g Cx TC C Cx k k n k C C Cx Cx k F = F f = F µ m g = m a o a = ( F µ m g) / m Tx ap CT ap k C T Tx Tx ap k C T 23/29

24 Example: Dumping a file cabinet (1) Steel on dry teel Free-body diagram A 50.0 kg teel file cabinet i in the back of a dump truck. The truck bed, alo made of teel, i lowly tilted. What i the ize of the tatic friction force when the truck bed i tilted by 20? At what angle will the file cabinet begin to lide? 24/29

25 Example: Dumping a file cabinet (2) F = winθ f = mginθ f = 0; x F = n wcoθ = n mgcoθ = 0; y f = mg θ = = 2 in (50.0kg)(9.80m/ )in N; File cabinet will begin to lide when: f = fmax = µ n= µ mgco θ; mg inθ f = mg inθ µ mg coθ = 0; inθ µ = = tan θ; θ = arctan µ = arctan(0.80) = 38.7 coθ 25/29

26 Friction, Car, & Antilock Brake The diagram how force acting on a car with front-wheel drive. Typically, F n > F n becaue the engine i over the front wheel. The larget frictional force f the tire can exert on the road i µ F n. Attempt to make the tire exert a force larger than thi caue the tire to burn rubber and actually reduce the force, ince µ k <µ. Note that while all point on the rolling tire have the ame peed v in the reference frame of the car, in the reference frame of the road the bottom of the tire i at ret, while top i moving forward with a peed of 2v. Antilock brake ene the wheel rotation and eae off if it cloe to topping, maintaining tatic friction with the road and allowing better control of teering than if the wheel were locked. 26/29

27 Antilock Brake 27/29

28 Example: The Effect of Antilock Brake A car i traveling at 30 m/ along a horizontal road. The coefficient of friction are µ =0.50 and µ k =0.40. (a) What i the braking ditance Δx a with antilock brake? (b) What i the braking ditance Δx b if the brake lock? v0 v = v0 + 2 aδ x and v= 0, o Δ x= 2a a = µ g and a = µ g a b k Δ v (30 m/) x = = = 91.7 m a µ g 2(0.5)(9.81 m/ ) 2 2 v0 (30 m/) 2 2µ k g 2(0.4)(9.81 m/ ) Δ x = = = m b 28/29