Understanding Friction, Drag, and Centripetal Force in University Physics Lectures

Lecture 10 Ch6. Friction, Drag, and Centripetal Force University Physics: Mechanics Dr.-Ing. Erwin Sitompul http://zitompul.wordpress.com 2014

Homework 9: Coin On A Book The figure below shows a coin of mass m at rest on a book that has been tilted at an angle with the horizontal. By experimenting, you find that when is increased to 13 , the coin is on the verge of sliding down the book, which means that even a slight increase beyond 13 produces sliding. What is the coefficient of static friction s between the coin and the book? Hint: Draw the free-body diagram of the coin first. 10/2 Erwin Sitompul University Physics: Mechanics

Solution of Homework 9: Coin On A Book Forces along the y axis: F ma = gcos F F gcos F F = cos F mg = net,y y = Why zero? 0 N N N Forces along the x axis: F ma = sin sin sin mg sin cos So, the coefficient of static friction is: tan13 = 0.231 = net,x x f = F mg = Why zero? 0 g F g F s = 0 s N = h d cos 0 s tan = = s s 10/3 Erwin Sitompul University Physics: Mechanics

Virtual Experiment: Determining k An object is kept in rest on an inclined surface. The angle is 26 , which is greater than the critical angle c ( s = tan c). Upon release, the object directly move and slide down to the bottom. It requires 4.29 s to reach the bottom, which is 18 m away from the initial point. Determine the coefficient of kinetic friction k between the object and the surface. 10/4 Erwin Sitompul University Physics: Mechanics

Example: Blue Block A block of mass m = 3 kg slides along a floor while a force F of magnitude 12 N is applied to it at an upward angle . The coefficient of kinetic friction between the block and the floor is k = 0.4. We can vary from 0 to 90 (with the block remains on the floor. What gives the maximum value of the block s acceleration magnitude a? 10/5 Erwin Sitompul University Physics: Mechanics

Example: Blue Block Forces along the y axis: F ma = F F + F mg = Forces along the x axis: F ma = x F f ma = cos F net,y net,x y x = sin 0 F N g F k y = F ma N k N F m F m What gives the maximum value of a? da/d = 0 = cos a sin g k 10/6 Erwin Sitompul University Physics: Mechanics

Example: Blue Block If a is given by F m F m = cos a sin g k then, the derivative of a with respect to is da d F m F m = + sin cos 0 k tan = = = = tan tan (0.4) 21.80 k 1 k 1 10/7 Erwin Sitompul University Physics: Mechanics

Example: Two Blocks Block B in the figure below weighs 711 N. The coefficient of static friction between block and table is 0.25; angle is 30 . Assume that the cord between B and the knot is horizontal. Find the maximum weight of block A for which the system will be stationary. 10/8 Erwin Sitompul University Physics: Mechanics

Example: Two Blocks TW Wall TB Knot TW TA FNB TA Block A fs,max TB TW Block B fs,max FgB FgA Knot FgA 10/9 Erwin Sitompul University Physics: Mechanics

Example: Two Blocks Forces along the y axis: 0 y F = 0 y F = sin T m g = TWy TW net, T W gA TWx fs,max W A Knot Forces along the x axis: 0 x F = 0 f = m g m g net, = s B A T sin W cos W FgA W s,max x = = cos T F = s B A W s NB sin cos cos T m g W s B = Btan W W A s = (0.25)(711)tan30 = 102.624 N 10/10 Erwin Sitompul University Physics: Mechanics

Example: Multiple Objects A block of mass m1 on a rough, horizontal surface is connected to a ball of mass m2 by a lightweight cord over a lightweight, frictionless pulley as shown in the figure below. A force of magnitude F at an angle with the horizontal is applied to the block as shown and the block slides to the right. The coefficient of kinetic friction between the block and surface is k. Find the magnitude of acceleration of the two objects. 10/11 Erwin Sitompul University Physics: Mechanics

Example: Multiple Objects Fy F FN Fx T m1 fk Forces in m1 F T f Fg1 = = m a m a T net, 1 1 x x x F k 1 = = cos F T F m a F k N T 1 Forces in m2 F = F = cos m a F 1 k N m2 m a m a net, 2 2 y y = = 0 0 F T net, F y g2 2 + y F F = + 2( ) T m g a N g1 Fg2 = = F F F m g F N g1 y sin F N 1 10/12 Erwin Sitompul University Physics: Mechanics

Example: Multiple Objects = = + 2( cos ) T T m g F a m a F 1 k N = sin F m g F N 1 + = ( ) cos ( sin ) m g a F m a m g F 2 1 k 1 + = + cos sin m a m a F F m g m g 1 2 k k 1 2 + = + sin ) ( + ( ) (cos ) m m a F m m g 1 2 k k 1 2 + + (cos sin ) ( m + ) F m m g = k k 1 2 a m 1 2 10/13 Erwin Sitompul University Physics: Mechanics

Example: Trio Blocks When the three blocks in the figure below are released from rest, they accelerate with a magnitude of 0.5 m/s2. Block 1 has mass M, block 2 has 2M, and block 3 has 2M. What is the coefficient of kinetic friction between block 2 and the table? 10/14 Erwin Sitompul University Physics: Mechanics

Example: Trio Blocks a Forces in m1 F = F = Forces in m2 F T = = m a Ma m a 2 Ma a net, 1 1 net, 2 2 y y x x a T T f 1 g1 2 1 k = + = + ( ) 2 T M g a T T F Ma 1 2 1 k N = = F m a 0 net, F 2 2 y y F FN T2 N g2 T1 = 2 F Mg N T1 T2 Forces in m3 F T F m1 m2 m3 fk = = m a 2 net, 3 3 M y y ( ) a Fg1 2 g3 = 2 ( ) T M g a Fg2 Fg3 2 10/15 Erwin Sitompul University Physics: Mechanics

Example: Trio Blocks = + ( = ) T T M g T a F 1 + 2 Ma ( ) ( = ) ( ) ) + = + 2 ( ) ( ) ) ( 2 2 M g ( a ) ( M g a Mg Ma 2 1 k N k = 2 F Mg N + k2 2 ( ) ( ) 2 Mg M g a M g a Ma = 2 ( ) T M g a 2 ( ) ( ) + 2 ( ) ( ) 2 M g a M g Mg a Ma = k 2 5 Mg Ma = k 2 Mg 5 a g g = 2 (9.8) 5(0.5) 2(9.8) 0.372 m s = = 2 10/16 Erwin Sitompul University Physics: Mechanics

The Drag Force and Terminal Speed A fluid is anything that can flow generally a gas or a liquid. When there is a relative velocity between a fluid and a body (either because the body moves through the fluid or because the fluid moves past the body), the body experiences a drag forceD that opposes the relative motion. Here we examine only cases in which air is the fluid, the body is blunt rather than slender, and the relative motion is fast enough so that the air becomes turbulent (breaks up into swirls) behind the body. In such cases, the magnitude of the drag force is related to the relative speed by an experimentally determined drag coefficient C according to : air specific density A : effective cross-sectional area of the body C : drag coefficient = C Av 2 D 1 2 10/17 Erwin Sitompul University Physics: Mechanics

The Drag Force and Terminal Speed 10/18 Erwin Sitompul University Physics: Mechanics

The Drag Force and Terminal Speed When a blunt body falls from rest through air, the drag force D is directed upward. This upward force D opposes the downward gravitational force Fg on the body. D F ma = g If the body falls long enough, D eventually equals Fg. This means that a= 0, and so the body s speed no longer increases. The body then falls at a constant speed, called the terminal speed vt. C Av = 2 t 0 F 1 2 g 2 F g = v t C A 10/19 Erwin Sitompul University Physics: Mechanics

The Drag Force and Terminal Speed Cyclists and downhill speed skiers try to maximize terminal speeds by reducing effective cross- sectional area 10/20 Erwin Sitompul University Physics: Mechanics

Example: Falling Cat If a falling cat reaches a first terminal speed of 97 km/h while it is wrapped up and then stretches out, doubling A, how fast is it falling when it reaches a new terminal speed 2 F g = = 95km h v ,1 t C A 2 F 95 g = = = v 67.18km h ,2 t 2 C A 2 10/21 Erwin Sitompul University Physics: Mechanics

Example: Raindrop A raindrop with radius R=1.5 mm falls from a cloud that is at height h=1200 m above the ground. The drag coefficient C for the drop is 0.6. Assume that the drop is spherical throughout its fall. The density of water w is 1000 kg/m3, and the density of air a is 1.2 kg/m3. (a) What is the terminal speed of the drop? (b) What would be the drop s speed just before impact if there were no drag force? 2 F 3 2( C ) 8 3 m g rg 2( ) 2( ) r g r w w v g A 4 3 ( (a) = = g = = = w w w v t 2 A C ) C A C C a w a a w a 3 8 (1000)(1.5 10 )(9.8) 3 (0.6)(1.2) = = = 7.38 m s 26.56 km h = = = = (b) 2 y 2 0, 2 ( g y ) 2(9.8)(0 1200) 153.36 m s v v y yv 0 y 552.10 km h 10/22 Erwin Sitompul University Physics: Mechanics

Homework 10A In the next figure, blocks A and B have weights of 44 N and 22 N, respectively. (a) Determine the minimum weight of block C to keep A from sliding if s, between A and the table is 0.20. (b) Block C suddenly is lifted off A. What is the acceleration of block A if k between A and the table is 0.15? 10/23 Erwin Sitompul University Physics: Mechanics

Homework 10B 1. The figure shows a 1.0-kg University Physics book connected to a 500-g tea mug. The book is pushed up the slope and reach a speed of 3.0 m/s before being released. The coefficients of friction are s = 0.50 and k = 0.20. (a)How far will the book slide upwards? (b)After the book reaches the highest point, will the book stick to the surface, or will it slide back down? 2. In downhill speed skiing, a skier is retarded by both the air drag force on the body and the kinetic frictional force on the skis. Suppose the slope angle is = 40.0 , the snow is dry with a coefficient of kinetic friction k = 0.04, the mass of the skier and equipment is m = 85.0 kg, the cross-sectional area of the (tucked) skier is A = 1.30 m2, the drag coefficient is C = 0.150, and the air density is 1.20 kg/m3. (a) What is the terminal speed? (b) If a skier can vary C by a slight amount dC by adjusting, say, the hand positions, what is the corresponding variation in the terminal speed? D fk 10/24 Erwin Sitompul University Physics: Mechanics