University Physics: Mechanics Ch6. Friction, Drag, and Centripetal Force Lecture 10 Dr.-Ing. Erwin Sitompul

10/2 Erwin SitompulUniversity Physics: Mechanics The figure below shows a coin of mass m at rest on a book that has been tilted at an angle θ with the horizontal. By experimenting, you find that when θ 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 μ s between the coin and the book? Hint: Draw the free-body diagram of the coin first. Homework 9: Coin On A Book

10/3 Erwin SitompulUniversity Physics: Mechanics Forces along the y axis: Forces along the x axis: Why zero? So, the coefficient of static friction is: Solution of Homework 9: Coin On A Book

10/4 Erwin SitompulUniversity Physics: Mechanics An object is kept in rest on an inclined surface. The angle θ is 26°, which is greater than the critical angle θ c (μ s = tanθ 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 μ k between the object and the surface. Virtual Experiment: Determining μ k θ 18 m

10/5 Erwin SitompulUniversity Physics: Mechanics 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 θ. The coefficient of kinetic friction between the block and the floor is μ k = 0.4. We can vary θ from 0 to 90° (with the block remains on the floor. What θ gives the maximum value of the block’s acceleration magnitude a? → Example: Blue Block

10/6 Erwin SitompulUniversity Physics: Mechanics Forces along the y axis:Forces along the x axis: What θ gives the maximum value of a? da/dθ = 0 Example: Blue Block

10/7 Erwin SitompulUniversity Physics: Mechanics then, the derivative of a with respect to θ is If a is given by Example: Blue Block

10/8 Erwin SitompulUniversity Physics: Mechanics Block B in the figure below weighs 711 N. The coefficient of static friction between block and table is 0.25; angle θ 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. Example: Two Blocks

10/9 Erwin SitompulUniversity Physics: Mechanics f s,max → Block B Block A Knot TBTB → TATA → TBTB → TATA → Wall TWTW → TWTW → F gA → Knot TWTW → f s,max → F gA → F NB → F gB → Example: Two Blocks

10/10 Erwin SitompulUniversity Physics: Mechanics Knot TWTW → f s,max → F gA → TWxTWx TWyTWy Forces along the y axis: Forces along the x axis: θ Example: Two Blocks

10/11 Erwin SitompulUniversity Physics: Mechanics A block of mass m 1 on a rough, horizontal surface is connected to a ball of mass m 2 by a lightweight cord over a lightweight, frictionless pulley as shown in the figure below. A force of magnitude F at an angle θ 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 μ k. Find the magnitude of acceleration of the two objects. Example: Multiple Objects

10/12 Erwin SitompulUniversity Physics: Mechanics fkfk → m1m1 m2m2 T → T → F g2 → FNFN → F g1 → F → FxFx FyFy θ Forces in m 2 Forces in m 1 Example: Multiple Objects

10/13 Erwin SitompulUniversity Physics: Mechanics Example: Multiple Objects

10/14 Erwin SitompulUniversity Physics: Mechanics When the three blocks in the figure below are released from rest, they accelerate with a magnitude of 0.5 m/s 2. 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? Example: Trio Blocks

10/15 Erwin SitompulUniversity Physics: Mechanics FNFN → fkfk → T1T1 → m1m1 T1T1 → F g1 → m2m2 F g2 → T2T2 → a a a m3m3 T2T2 → F g3 → Forces in m 1 Forces in m 3 Forces in m 2 Example: Trio Blocks

10/16 Erwin SitompulUniversity Physics: Mechanics Example: Trio Blocks

10/17 Erwin SitompulUniversity 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 force D 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 ρ : air specific density A : effective cross-sectional area of the body C : drag coefficient

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

10/19 Erwin SitompulUniversity 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 F g on the body. → →  If the body falls long enough, D eventually equals F g. 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 v t.

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

10/21 Erwin SitompulUniversity Physics: Mechanics Drag Friction Accident  Space Shuttle Columbia disintegrated in the air on , killing all seven crew members.  During the launch, a piece of foam insulation broke off from the external tank and struck the left wing.  When the shuttle reentered the atmosphere, the damage allowed hot atmospheric gases to penetrate and destroy the internal wing structure

10/22 Erwin SitompulUniversity Physics: Mechanics 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 Example: Falling Cat

10/23 Erwin SitompulUniversity Physics: Mechanics 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 ρ w is 1000 kg/m 3, and the density of air ρ a is 1.2 kg/m 3. (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? Example: Raindrop (a) (b)

10/24 Erwin SitompulUniversity Physics: Mechanics Homework 10A 1.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 μ s, between A and the table is (b)Block C suddenly is lifted off A. What is the acceleration of block A if μ k between A and the table is 0.15? 2.Compare the drag force on an airplane when it flies at an altitidue of m with 1000 km/h and when it files at m with 500 km/h. The density of air is 0.38 kg/m 3 at the first case and 0.67 kg/m 3 at the second case.

10/25 Erwin SitompulUniversity 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 μ s = 0.50 and μ k = (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 θ = 40.0°, the snow is dry with a coefficient of kinetic friction μ 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 m 2, the drag coefficient is C = 0.150, and the air density is 1.20 kg/m 3. (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 fkfk