Section 8.1 Characteristics of Dry

Engineering Mechanics: Statics Fourteenth Edition Chapter 8 Friction Section 8.1 Characteristics of Dry Friction Section 8.2 Problems Involving Dry Friction Section 8.3 Wedges Section 8.5 Frictional Forces on Flat Belts Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Section 8.1: Characteristics of Dry Friction (1 of 4) Friction is defined as a force of resistance acting on a body which prevents or resists the slipping of a body relative to a second body. Experiments show that frictional forces act tangent (parallel) to the contacting surface in a direction opposing the relative motion or tendency for motion. For the body shown in the figure to be in equilibrium, the following must be true: Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Section 8.1: Characteristics of Dry Friction (2 of 4) To study the characteristics of the friction force F, let us assume that tipping does not occur (i.e., “h” is small or “a” is large). Then we gradually increase the magnitude of the force P. Typically, experiments show that the friction force F varies with P, as shown in the right figure above. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Section 8.1: Characteristics of Dry Friction (3 of 4) The maximum friction force is attained just before the block begins to move (a situation that is called “impending motion”). The value of the force is found using 𝐹𝑠 = µ𝑠 𝑁 Where is called the coefficient of static friction. The value of depends on the two in contact. Once the block begins to move, the frictional force typically drops and is given by 𝐹𝑘 = µ𝑘 𝑁 The value of (coefficient of kinetic friction) is less than Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Section 8.1: Characteristics of Dry Friction (4 of 4) It is also very important to note that the friction force may be less than the maximum friction force. So, just because the object is not moving, don’t assume the friction force is at its maximum of F= 𝜇𝑠 𝑁 unless you are told or know motion is impending! Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Determing mu sub s Experimentally A block with weight w is placed on an inclined plane. The plane is slowly tilted until the block just begins to slip. The inclination, is noted. Analysis of the block just before it begins to move gives (using F= 𝜇𝑠 𝑁) Using these two equations, we get This simple experiment allows us to find the between two materials in contact. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Section 8.2: Problems Involving Dry Friction Steps for solving equilibrium problems involving dry friction: 1. Draw necessary free body diagrams. Make sure that you show the friction force in the correct direction (it always opposes the motion or impending motion). 2. Determine the number of unknowns. Do not assume that F= 𝜇𝑠 𝑁 unless the impending motion condition is given. 3. Apply the equations of equilibrium and appropriate frictional equations to solve for the unknowns. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Impending Tipping Versus Slipping (1 of 2) For a given W and h of the box, how can we determine if the block will slide or tip first? In this case, we have four unknowns (F, N, x, and P) and only the three E-of-E. Hence, we have to make an assumption to give us another equation (the friction equation!). Then we can solve for the unknowns using the three E-of-E. Finally, we need to check if our assumption was correct. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Impending Tipping Versus Slipping (2 of 2) Assume: Slipping occurs Known: F= 𝜇𝑠 𝑁 Solve: x, P, and N Check: Or Assume: Tipping occurs Known: Solve: P, N, and F Check: F≤ 𝜇𝑠 𝑁 Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Analysis of a Wedge A wedge is a simple machine in which a small force P is used to lift a large weight W. To determine the force required to push the wedge in or out, it is necessary to draw FBDs of the wedge and the object on top of it. It is easier to start with a FBD of the wedge since you know the direction of its motion. Note that: a. the friction forces are always in the direction opposite to the motion, or impending motion, of the wedge; b. the friction forces are along the contacting surfaces; and, c. the normal forces are perpendicular to the contacting surfaces. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Analysis of a Wedge Next, a FBD of the object on top of the wedge is drawn. Please note that: a. at the contacting surfaces between the wedge and the object, the forces are equal in magnitude and opposite in direction to those on the wedge; and, b. all other forces acting on the object should be shown. To determine the unknowns, we must apply E-of-E, and  Fx = 0  Fy = 0 to the wedge and the object as well as the F = S N. impending motion frictional equation, Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Analysis of a Wedge Note: If the object is to be lowered, then the wedge needs to be pulled out. If the value of the force P needed to remove the wedge is positive, then the wedge is self-locking, i.e., it will not come out on its own. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Belt Analysis Consider a flat belt passing over a fixed curved surface with the total angle of contact equal to  radians. If the belt slips or is just about to slip, then T2 must be larger than T1 and the motion resisting friction forces. Hence, T must be greater than T1. 2 Detailed analysis (please refer to your textbook) shows that T2 = T1 e   where  is the coefficient of static friction between the belt and the surface. Be sure to use radians when using this formula!! Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Group Problem Solving (1 of 3) Given: Automobile has a mass of 2000 k g and ilo ram Find: The smallest magnitude of F required to move the car if the back brakes are locked and the front wheels are free to roll. Plan: a. Draw FBDs of the car. b. Determine the unknowns. c. Apply the E-of-E and friction equations to solve for the unknowns. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Group Problem Solving (2 of 3) Here is the correct FBD: Note that there are four unknowns: Equations of Equilibrium: Assume that the rear wheels are on the verge of slip. Thus Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Group Problem Solving (3 of 3) Solving Equations (1) to (4), F = 2762 N and Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Example 1 (1 of 3) Given: The 3000-lb load is applied to wedge B. The coefficient of static friction between A and C and between B and D is 0.3, and between A and B it is 0.4. Assume the wedges have negligible weight. Find: The smallest force P needed to lift 3000 lb load. Plan: 1. Draw FBDs of wedge A and wedge B. 2. Apply the E-of-E to wedge B. Why do wedge B first? 3. Apply the E-of-E to wedge A. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Example 1 (2 of 3) Applying the E-of-E to wedge B, we get → +  FX = N sin 15 + 0.4 N cos 15 − ND = 0  +  FY = N cos 15 − 0.4 N sin 15 − 0.3 ND − 3000 = 0 Solving these two equations, we get N = 4485 lb, ND = 2894 lb Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Example 1 (3 of 3) Applying the E-of-E to wedge A, we get  +  FY = NC + 0.4 (4485) sin 15 − 4485 cos 15 = 0; NC = 3868 lb → +  FX = P − 0.3 (3868) − 4485 sin 15 − 1794 cos 15 = 0; P = 4054 lb Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Attention Quiz (1 of 2) 1. A 10 lb block is in equilibrium. What is the magnitude of the friction force between this block and the surface? A. 0 lb B. 1 lb C. 2 lb D. 3 lb Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Attention Quiz (2 of 2) 2. The ladder AB is positioned as shown. What is the direction of the friction force on the ladder at B. A. B. C. D. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Reading Quiz (1 of 2) 1. A friction force always acts Fill in the blank to the contact surface. A. Normal B. At 45° C. Parallel D. At the angle of static friction Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Reading Quiz (2 of 2) 2. If a block is stationary, then the friction force acting on it is Fill in the blank . A. B. C. D. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Concept Quiz (1 of 2) 1. A 100 lb box with a wide base is pulled by a force P and Which force orientation requires the least force to begin sliding? A. P(A) B. P(B) C. P(C) D. Can not be determined Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Concept Quiz (2 of 2) 2. A ladder is positioned as shown. Please indicate the direction of the friction force on the ladder at B. A. B. C. D. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved