Chapte 5 Fiction When an object is in motion it is usually in contact with a viscous mateial (wate o ai) o some othe suface. So fa, we have assumed that moving objects don t inteact with thei suoundings but usually they do inteact and this inteaction opposes the motion of the object. This esistance to motion is called that foce of fiction. Have you eve tied to push something heavy, like a table o bookcase, acoss the floo? If you have you pobably noticed that you have to push eally had to stat the object moving and then, once it stats moving, you don t have to push as had to keep it moving. Imagine that we have the following: m f F The foce f is the fiction foce. If ou object is not moving: x F = F f = 0 F= f Since ou object is not moving, this fictional foce is called the foce of f. This foce aises fom the fact that while sufaces static fiction ( s ) may feel vey smooth, thee ae potusions coming out fom the suface that can be seen on the micoscopic level. As you attempt to push the object to the ight with F, these potusions exet the same foce on the floo. Fom Newton s 3 d law we know that the floo will exet an equal and opposite foce on the object. As we continue to incease the applied foce, the fictional foce will incease. Eventually we will apply enough foce so that we exceed the maximum possible esistance. The maximum possible value fo the foce of static fiction is: Page 1
fs, max = µ sn (1) F is the nomal foce and µ n s is the coefficient of static fiction. In geneal: f s µ n As the block begins to move, it seems that you don t need to push as had to keep the object moving as you did to stat it moving. Once ou object is in motion, we now have the foce of kinetic fiction. The foce of kinetic fiction is: s fk = µ kn µ k = coefficient of kinetic fiction () We don t have to push as had because: fk < fs, max =µ sn Imagine that a hockey puck on a fozen pond is given an initial speed of 0.0 m/s. If the puck slides 115 m befoe coming to est, what is the coefficient of kinetic fiction? 1. Daw a fee body diagam: n f k mg 1. Apply Newton s nd law: y Fx = fk = ma (3) x F = n mg = 0 n = mg (4) Page
. Solve fo the acceleation in tems of the coefficient of kinetic fiction: Substitute (5) into equation (3): fk =µ kn =µ kmg (5) µ kmg = max ax = µ kg 3. Now apply kinematics to kind the coefficient of fiction: m m v xi = 0.0, v xf = 0, x = 115 m s s v = v + a x xf xi x ( ) 0= v + µ g x xi k m 0.0 v xi µ = = s k = 0.177 g x m 9.8 ( 115 m) s Example 1: A 3.00 kg block stats at est at the top of a 30 o incline and slides a distance of.00 m down the incline in 1.50 s. Find: a) the magnitude of the acceleation of the block, b) the coefficient of kinetic fiction between the block and plane, c) the fiction foce acting on the block, and d) the speed of the block afte it slid.00 m. Page 3
Example : A block of mass m1 =.5 kg is at est on the inclined plane shown below. The coefficient of kinetic fiction between the block and the plane is 0.100. The block is attached to a second block of mass m =.0 kg that hangs feely by a sting that passes ove a fictionless, massless pulley (the blocks will now stat to move). What is the speed of the system when the second block has fallen 30.0 cm? Page 4
Example 3: A toy chest and its contents have a combined weight of 180 N. The coefficient of static fiction between toy chest and floo is 0.4. The child attempts to move the chest acoss the floo by pulling on an attached ope. A. If is 4, what is the magnitude of the foce that the child must exet on the ope to put the chest on the vege of moving? B. Wite an expession fo the magnitude F equied to put the chest on the vege of moving as a function of the angle. C. Detemine the value of fo which F is a minimum and that minimum magnitude. Page 5
Example 4: A loaded penguin sled weighing 80 N ests on a plane inclined at angle θ= 0 to the hoizontal. Between the sled and the plane, the coefficient of static fiction is 0.5, and the coefficient of kinetic fiction is 0.15. A. What is the least magnitude of the foce F paallel to the plane, that will pevent the sled fom slipping down the plane? B. What is the minimum magnitude F that will stat the sled moving up the plane? C. What value of F is equied to move the sled up the plane at constant velocity? Page 6
Unifom Cicula Motion Recall that when a paticle moves along a cicula path of adius, it expeiences a adial acceleation of magnitude: a = v We have aleady seen that a always points towad the cente of the cicula path and is always othogonal to the velocity vecto. Conside a ball of mass m that is tied to a sting of length that is being whiled at a constant speed in a hoizontal cicula path. The inetia of the ball causes it to want to move in a staight-line path, but the sting pevents it fom moving along that path. v F If we apply Newton s nd law along the adial diection: F mv = ma = (6) Theefoe, a foce causing the centipetal acceleation acts towad the cente of the cicula path. Without that foce, ou ball would move off in a staight-line. Imagine that we have a canonical pendulum of length L. mass m. that makes an angle θ with the vetical: Page 7
The fee-body diagam fo the mass is: Tcosθ T sinθ mg Applying Newton s nd law to the mass: y F = Tcosθ mg = 0 Tcosθ = mg mv F = Tsinθ= mg mv sinθ= cosθ Lsinθ v g tanθ = Lsinθ v = Lg sinθ tanθ Page 8
Example 5: A 1500 kg ca is moving on a flat, hoizontal oad negotiates a cuve. If the adius of the cuve is 35.0 m and the coefficient of static fiction between the ties and the dy pavement is 0.500, find the maximum speed the ca can have and still make the tun successfully. Page 9
Example 6: A cate of eggs is located in the middle of a flatbed pickup tuck as the tuck negotiates an unbanked cuve in the oad. The cuve may be egaded as an ac of a cicle of adius 35.0 m. If the coefficient of static fiction between the cate and the tuck is 0.600, how fast can the tuck be moving without the cate sliding? Page 10
Example 7: A civil enginee wishes to design a cuved exit amp fo a highway in such a way that a ca will not have to ely on fiction to ound the cuve without skidding. In othe wods, a ca moving at a designated speed can negotiate the cuve even when the oad is coveed with ice. Such a amp is usually banked; this means that the oadway is tilted towad the inside of the cuve. Suppose that the designated speed of the amp is to be 13.4 m/s (30 mi/h) and the adius of the cuve is 50.0 m. At what angle should the cuve be banked? Page 11
Example 8: A ide at an amusement pak caies people in a vetical cicle at a constant speed such that the nomal foce exeted by the seats is always inwad towad the cente of the cicle. At the top, the nomal foce exeted by the seat equals the peson s weight, mg. What is the foce exeted by the seat at the bottom of the loop? Page 1