# Sections 8.3, 8.4, & 8.5 - cbafaculty.org WEDGES AND FRICTIONAL FORCES ON FLAT BELTS Todays Objectives: Students will be able to: In-Class Activities: a) Determine the forces on a wedge. Check Homework, if any b) Determine tension in a belt. Reading Quiz Applications Analysis of a Wedge Analysis of a Belt Concept Quiz Group Problem Solving Attention Quiz Statics, Fourteenth Edition R.C. Hibbeler

Copyright 2016 by Pearson Education, Inc. All rights reserved. READING QUIZ 1. A wedge allows a ______ force P to lift a _________ weight W. A) (large, large) W B) (small, large) C) (small, small) D) (large, small) 2. Considering friction forces and the indicated motion of the belt, how are belt tensions T1 and T2 related? A) T1 > T2 B) T1 = T2 C) T1 < T2 D) T1 = T2 e

Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. APPLICATIONS Wedges are used to adjust the elevation or provide stability for heavy objects such as this large steel pipe. How can we determine the force required to pull the wedge out? When there are no applied forces on the wedge, will it stay in place (i.e., be self-locking) or will it come out on its own? Under what physical conditions will it come out? Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved.

APPLICATIONS (continued) Belt drives are commonly used for transmitting the torque developed by a motor to a wheel attached to a pump, fan or blower. How can we decide if the belts will function properly, i.e., without slipping or breaking? Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. APPLICATIONS (continued) In the design of a band brake, it is essential to analyze the frictional forces acting on the band (which acts like a belt). How can you determine the tension in the cable pulling on the

band? Also from a design perspective, how are the belt tension, the applied force P and the torque M, related? Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. ANALYSIS OF A WEDGE W 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. Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. ANALYSIS OF A WEDGE (continued) 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, Fx = 0 and Fy = 0, to the wedge and the object as well as the impending motion frictional equation, F = S N.

Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. ANALYSIS OF A WEDGE (continued) Now of the two FBDs, which one should we start analyzing first? We should start analyzing the FBD in which the number of unknowns are less than or equal to the number of E-of-E and frictional equations. Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. ANALYSIS OF A WEDGE (continued) 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. Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by 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, T2 must be greater than T1. 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!! Statics, Fourteenth Edition R.C. Hibbeler

Copyright 2016 by Pearson Education, Inc. All rights reserved. EXAMPLE 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. Plan: Find: The smallest force P needed to lift 3000 lb load. 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. Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved.

EXAMPLE (continued) N 15 3000 lb F = 0.4 N P A 15 15 F=0.4 N 0.3NC NC FBD of Wedge A

B N 15 ND FD=0.3ND FBD of Wedge B 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 Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. EXAMPLE (continued) N = 4485 lb

15 3000 lb F = 0.4(4485) lb P A 15 15 0.3NC NC FBD of Wedge A F=0.4N N ND

B 15 FD=0.3ND FBD of Wedge B Applying the E-of-E to wedge A, we get + FY = NC + 0.4(4485) sin15 4485 cos15 = 0; NC = 3868 lb + FX = P 0.3(3868) 4485 sin15 1794 cos15 = 0 ; P = 4054 lb Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. CONCEPT QUIZ 1. Determine the direction of the friction force on object B at the contact point between A and B.

A) B) C) D) 2. The boy (hanging) in the picture weighs 100 lb and the woman weighs 150 lb. The coefficient of static friction between her shoes and the ground is 0.6. The boy will ______ ? A) Be lifted up B) Slide down C) Not be lifted up D) Not slide down Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. GROUP PROBLEM SOLVING Given: A force P is applied to move

wedge A to the right. The spring is compressed a distance of 175 mm. The static friction coefficient is S = 0.35 for all contacting surfaces. Neglect the weight of A and B. Find: The smallest force P needed to move wedge A. Plan: Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. GROUP PROBLEM SOLVING (continued) Plan: 1. Draw FBDs of block B and wedge A. 2. Apply the E-of-E to block B to find the friction force when the wedge is on the verge of moving. 3. Apply the E-of-E to wedge A to find the smallest force

needed to cause sliding. Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. GROUP PROBLEM SOLVING (continued) FSP = 15(0.175) = 2.625 kN NC NB P FB=0.35NB 10 FA=0.35NA NB 10 FB=0.35NB

FBD of Block B NA FBD of Wedge A Using the spring formula: Fsp = K x = (15 kN/m) (0.175m) = 2.625 kN If the wedge is on the verge of moving to the right, then slipping will have to occur at both contact surfaces. Thus, FA = S NA = 0.35 NA and FB = 0.35 NB. Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. GROUP PROBLEM SOLVING (continued) FSP = 2.625 kN NC NB P

FB=0.35NB 10 FA=0.35NA NB 10 FB=0.35NB FBD of Block B NA FBD of Wedge A Applying the E-of-E to the Block B, we get: + FY = NB 2.625 = 0 NB = 2.625 kN Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc.

All rights reserved. GROUP PROBLEM SOLVING (continued) FSP = 2.625 kN NC 2.625 kN P FB=0.35(2.625) 10 FA=0.35NA NB FB=0.35NB FBD of Block B 10 NA FBD of Wedge A

Applying the E-of-E to Wedge A: + FY = NA cos 10 0.35NA sin 10 2.625 = 0 NA = 2.841 kN + FX = P 0.35(2.625) 0.35(2.841) cos10 2.841 sin10 = 0 P = 2.39 kN Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. ATTENTION QUIZ 1. When determining the force P needed to lift the block of weight W, it is easier to draw a FBD of ______ first. A) The wedge W B) The block C) The horizontal ground D) The vertical wall 2. In the analysis of frictional forces on a flat belt, T2 = T1 e .

In this equation, equals ______ . A) Angle of contact in degrees B) Angle of contact in radians C) Coefficient of static friction D) Coefficient of kinetic friction Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved. End of the Lecture Let Learning Continue Statics, Fourteenth Edition R.C. Hibbeler Copyright 2016 by Pearson Education, Inc. All rights reserved.