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A body has improper supports if it will not remain in equilibrium under the action of the loads exerted on it. The body with improper supports will move when the loads are applied. 5. Dry Friction If a body rests on an inclined plane, the friction force exerted on it by the surface prevents it from sliding down the incline. The question is: what is the steepest incline on which the body can rest? A body is placed on a horizontal surface. The body is pushed with a small horizontal force F . If the force F is suf®ciently small, the body does not move. Figure 5.1 shows the free-body diagram of the body, where the force W is the weight of the body, and N is the normal force exerted by the surface. The force F is the horizontal force, and F f is the friction force exerted by the surface. Friction force arises in part from the interactions of the roughness, or asperities, of the contacting surfaces. The body is in equili- brium and F f F . The force F is slowly increased. As long as the body remains in equilibrium, the friction force F f must increase correspondingly, since it equals the force F . The body slips on the surface. The friction force, after reaching the maximum value, cannot maintain the body in equilibrium. The force applied to keep the body moving on the surface is smaller than the force required to cause it to slip. The fact that more force is required to start the body sliding on a surface than to keep it sliding is explained in part by the necessity to break the asperities of the contacting surfaces before sliding can begin. The theory of dry friction, or Coloumb friction, predicts: j The maximum friction forces that can be exerted by dry, contacting surfaces that are stationary relative to each other j The friction forces exerted by the surfaces when they are in relative motion, or sliding Figure 5.1 46 Statics Statics 5.1 Static Coef®cient of Friction The magnitude of the maximum friction force, F f , that can be exerted between two plane dry surfaces in contact is F f m s N Y 5X1 where m s is a constant, the static coef®cient of friction, and N is the normal component of the contact force between the surfaces. The value of the static coef®cient of friction, m s , depends on: j The materials of the contacting surfaces j The conditions of the contacting surfaces: smoothness and degree of contamination Typical values of m s for various materials are shown in Table 5.1. Equation (5.1) gives the maximum friction force that the two surfaces can exert without causing it to slip. If the static coef®cient of friction m s between the body and the surface is known, the largest value of F one can apply to the body without causing it to slip is F F f m s W . Equation (5.1) deter- mines the magnitude of the maximum friction force but not its direction. The friction force resists the impending motion. 5.2 Kinetic Coef®cient of Friction The magnitude of the friction force between two plane dry contacting surfaces that are in motion relative to each other is F f m k N Y 5X2 where m k is the kinetic coef®cient of friction and N is the normal force between the surfaces. The value of the kinetic coef®cient of friction is generally smaller than the value of the static coef®cient of friction, m s . Table 5.1 Typical Values of the Static Coef®cient of Friction Materials mm s Metal on metal 0.15±0.20 Metal on wood 0.20±0.60 Metal on masonry 0.30±0.70 Wood on wood 0.25±0.50 Masonry on masonry 0.60±0.70 Rubber on concrete 0.50±0.90 5. Dry Friction 47 Statics To keep the body in Fig. 5.1 in uniform motion (sliding on the surface) the force exerted must be F F f m k W . The friction force resists the relative motion, when two surfaces are sliding relative to each other. The body RB shown in Fig. 5.2a is moving on the ®xed surface 0. The direction of motion of RB is the positive axis x. The friction force on the body RB acts in the direction opposite to its motion, and the friction force on the ®xed surface is in the opposite direction (Fig. 5.2b). 5.3 Angles of Friction The angle of friction, y, is the angle between the friction force, F f jF f j, and the normal force, N jNj, to the surface (Fig. 5.3). The magnitudes of the normal force and friction force and that of y are related by F f R sin y N R cos yY where R jRjjN F f j. The value of the angle of friction when slip is impending is called the static angle of friction, y s : tan y s m s X Figure 5.2 48 Statics Statics The value of the angle of friction when the surfaces are sliding relative to each other is called the kinetic angle of friction, y k : tan y k m k X References 1. A. Bedford and W. Fowler, Dynamics. Addison Wesley, Menlo Park, CA, 1999. 2. A. Bedford and W. Fowler, Statics. Addison Wesley, Menlo Park, CA, 1999. 3. F. P. Beer and E. R. Johnston, Jr., Vector Mechanics for Engineers: Statics and Dynamics. McGraw-Hill, New York, 1996. 4. R. C. Hibbeler, Engineering Mechanics: Statics and Dynamics. Prentice-Hall, Upper Saddle River, NJ, 1995. 5. T. R. Kane, Analytical Elements of Mechanics, Vol. 1. Academic Press, New York, 1959. 6. T. R. Kane, Analytical Elements of Mechanics, Vol. 2. Academic Press, New York, 1961. 7. T. R. Kane and D. A. Levinson, Dynamics. McGraw-Hill, New York, 1985. 8. D. J. McGill and W. W. King, Engineering Mechanics: Statics and an Introduction to Dynamics. PWS Publishing Company, Boston, 1995. 9. R. L. Norton, Machine Design. Prentice-Hall, Upper Saddle River, NJ, 1996. 10. R. L. Norton, Design of Machinery. McGraw-Hill, New York, 1999. 11. W. F. Riley and L. D. Sturges, Engineering Mechanics: Statics. John Wiley & Sons, New York, 1993. 12. I. H. Shames, Engineering Mechanics: Statics and Dynamics. Prentice-Hall, Upper Saddle River, NJ, 1997. 13. R. W. Soutas-Little and D. J. Inman, Engineering Mechanics: Statics. Prentice- Hall, Upper Saddle River, NJ, 1999. Figure 5.3 References 49 Statics 2 Dynamics DAN B. MARGHITU, BOGDAN O. CIOCIRLAN, AND CRISTIAN I. DIACONESCU Department of Mechanical Engineering, Auburn University, Auburn, Alabama 36849 Inside 1. Fundamentals 52 1.1 Space and Time 52 1.2 Numbers 52 1.3 Angular Units 53 2. Kinematics of a Point 54 2.1 Position, Velocity, and Acceleration of a Point 54 2.2 Angular Motion of a Line 55 2.3 Rotating Unit Vector 56 2.4 Straight Line Motion 57 2.5 Curvilinear Motion 58 2.6 Normal and Tangential Components 59 2.7 Relative Motion 73 3. Dynamics of a Particle 74 3.1 Newton's Second Law 74 3.2 Newtonian Gravitation 75 3.3 Inertial Reference Frames 75 3.4 Cartesian Coordinates 76 3.5 Normal and Tangential Components 77 3.6 Polar and Cylindrical Coordinates 78 3.7 Principle of Work and Energy 80 3.8 Work and Power 81 3.9 Conservation of Energy 84 3.10 Conservative Forces 85 3.11 Principle of Impulse and Momentum 87 3.12 Conservation of Linear Momentum 89 3.13 Impact 90 3.14 Principle of Angular Impulse and Momentum 94 4. Planar Kinematics of a Rigid Body 95 4.1 Types of Motion 95 4.2 Rotation about a Fixed Axis 96 4.3 Relative Velocity of Two Points of the Rigid Body 97 4.4 Angular Velocity Vector of a Rigid Body 98 4.5 Instantaneous Center 100 51 1. Fundamentals 1.1 Space and Time S pace is the three-dimensional universe. The distance between two points in space is the length of the straight line joining them. The unit of length in the International System of units, or SI units, is the meter (m). In U.S. customary units, the unit of length is the foot (ft). The U.S. customary units use the mile (mi) 1 mi 5280 ft and the inch (in) 1 ft 12 in. The time is a scalar and is measured by the intervals between repeatable events. The unit of time is the second (s) in both SI units and U.S. customary units. The minute (min), hour (hr), and day are also used. The velocity of a point in space relative to some reference is the rate of change of its position with time. The velocity is expressed in meters per second (mas) in SI units, and is expressed in feet per second (ftas) in U.S. customary units. The acceleration of a point in space relative to some reference is the rate of change of its velocity with time. The acceleration is expressed in meters per second squared mas 2 in SI units, and is expressed in feet per second squared ftas 2 in U.S. customary units. 1.2 Numbers Engineering measurements, calculations, and results are expressed in numbers. Signi®cant digits are the number of meaningful digits in a number, counting to the right starting with the ®rst nonzero digit. Numbers can be rounded off to a certain number of signi®cant digits. The value of p can be expressed to three signi®cant digits, 3.14, or can be expressed to six signi®cant digits, 3.14159. The multiples of units are indicated by pre®xes. The common pre®xes, their abbreviations, and the multiples they represent are shown in Table 1.1. For example, 5 km is 5 kilometers, which is 5000 m. 4.6 Relative Acceleration of Two Points of the Rigid Body 102 4.7 Motion of a Point That Moves Relative to a Rigid Body 103 5. Dynamics of a Rigid Body 111 5.1 Equation of Motion for the Center of Mass 111 5.2 Angular Momentum Principle for a System of Particles 113 5.3 Equations of Motion for General Planar Motion 115 5.4 D'Alembert's Principle 117 References 117 52 Dynamics Dynamics Some useful unit conversions are presented in Table 1.2. For example, 1miahr in terms of ftas is (1 mi equals 5280 ft and 1 hr equals 3600 s) 1 mi hr 1 1mi 5280 ft 1hr 3600 s 1 5280 ft 3600 s 1X47 ft s X 1.3 Angular Units Angles are expressed in radians (rad) in both SI and U.S. customary units. The value of an angle y in radians (Fig. 1.1) is the ratio of the part of the Table 1.1 Pre®xes Used in SI Units Pre®x Abbreviation Multiple nano- n 10 À9 micro- m 10 À6 mili- m 10 À3 kilo- k 10 3 mega- M 10 6 giga- G 10 9 Table 1.2 Unit Conversions Time 1 minute 60 seconds 1 hour 60 minutes 1 day 24 hours Length 1 foot 12 inches 1 mile 5280 feet 1 inch 25.4 millimeters 1 foot 0.3048 meter Angle 2p radians 360 degrees Figure 1.1 1. Fundamentals 53 Dynamics circumference s subtended by y to the radius R of the circle, y s R X Angles are also expressed in degrees. There are 360 degrees 360 in a complete circle. The complete circumference of the circle is 2pR. Therefore, 360 2p radX 2. Kinematics of a Point 2.1 Position, Velocity, and Acceleration of a Point One may observe students and objects in a classroom, and their positions relative to the room. Some students may be in the front of the classroom, some in the middle of the classroom, and some in the back of the classroom. The classroom is the ``frame of reference.'' One can introduce a cartesian coordinate system xY yY z with its axes aligned with the walls of the class- room. A reference frame is a coordinate system used for specifying the positions of points and objects. The position of a point P relative to a given reference frame with origin O is given by the position vector r from point O to point P (Fig. 2.1). If the point P is in motion relative to the reference frame, the position vector r is a function of time t (Fig. 2.1) and can be expressed as r rtX The velocity of the point P relative to the reference frame at time t is de®ned by v dr dt r lim Dt30 rt DtÀrt Dt Y 2X1 Figure 2.1 54 Dynamics Dynamics where the vector rt DtÀrt is the change in position, or displacement of P Y during the interval of time Dt (Fig. 2.1). The velocity is the rate of change of the position of the point P . The magnitude of the velocity v is the speed v jvj. The dimensions of v are (distance)a(time). The position and velocity of a point can be speci®ed only relative to a reference frame. The acceleration of the point P relative to the given reference frame at time t is de®ned by a dv dt v lim Dt30 vt DtÀvt Dt Y 2X2 where vt DtÀvt is the change in the velocity of P during the interval of time Dt (Fig. 2.1). The acceleration is the rate of change of the velocity of P at time t (the second time derivative of the displacement), and its dimensions are (distance)a(time) 2 . 2.2 Angular Motion of a Line The angular motion of the line L, in a plane, relative to a reference line L 0 ,in the plane, is given by an angle y (Fig. 2.2). The angular velocity of L relative to L 0 is de®ned by o dy dt yY 2X3 and the angular acceleration of L relative to L 0 is de®ned by a do dt d 2 y dt 2 o yX 2X4 The dimensions of the angular position, angular velocity, and angular acceleration are (rad), (radas), and radas 2 , respectively. The scalar coordi- Figure 2.2 2. Kinematics of a Point 55 Dynamics nate y can be positive or negative. The counterclockwise (ccw) direction is considered positive. 2.3 Rotating Unit Vector The angular motion of a unit vector u in a plane can be described as the angular motion of a line. The direction of u relative to a reference line L 0 is speci®ed by the angle y in Fig. 2.3a, and the rate of rotation of u relative to L 0 is de®ned by the angular velocity o dy dt X The time derivative of u is speci®ed by du dt lim Dt30 ut DtÀut Dt X Figure 2.3a shows the vector u at time t and at time t Dt. The change in u during this interval is Du ut Dtut , and the angle through which u Figure 2.3 56 Dynamics Dynamics [...]... ay uy Y 2X63 2 d 2r dy d 2r 2 À r o2 r À r o2 ar 2 À r dt dt dt d 2y dr d y dr r a 2o r a 2 oX r ay r 2 2 dt dt dt dt 2X64 where The term a is called the angular acceleration d 2y y dt 2 Dynamics where o d yadt is the angular velocity If we substitute Eq (2. 57) into Eq (2. 56), the velocity of P is 72 Dynamics The radial component of the acceleration Àr o2 is called... v2 r2 2 1 1 1 m d v2 mv2 À mv2 Y F Á dr 3X20 2 2 2 1 r1 v2 2 1 where v1 and v2 are the magnitudes of the velocity at the positions r1 and r2 The kinetic energy of a particle of mass m with the velocity v is the term 1 1 T mv Á v mv2 Y 2 2 3X21 where jvj v The work done as the particle moves from position r1 to position r2 is r2 F Á d rX 3X 22 U 12 r1 The principle of work and... dt dt dt dt y d 2y d y d uy X r 2 uy r dt dt dt 2X61 As P moves, uy also rotates with angular velocity d yadt The time derivative of the unit vector uy is in the Àur direction if d yadt is positive: d uy dy À u X dt dt r 2X 62 If Eq (2. 62) and Eq (2. 57) are substituted into Eq (2. 61), the acceleration of the point P is 4 2 5 ! d 2r dy d 2y dr d y Àr u X ur r 2 2 a dt 2 dt dt dt dt y... y aj z akX 2X 42 The unit tangent vector is t d r d a rH a H Y d a ds s a 2X43 where a prime denotes differentiation with respect to a and rH x H i y H j z H kX Using the fact that jtj 1, one may write s H rH Á rH 1a2 x H 2 y H 2 z H 2 1a2 X The arc length s may be computed with the relation a s x H 2 y H 2 z H 2 1a2 d aY ao 2X44 2X45 where ao... from Eq (2. 44) may be substituted into Eq (2. 43) to calculate the tangent vector x Hi y Hj z Hk t q X x H 2 y H 2 z H 2 From Eqs (2. 43) and (2. 28), the normal vector is dt d t d a r rHH rH s HH r r À rHH s H À rH s HH X nr ds d a ds s H s H s H 2 s H 3 2X46 2X47 69 2 Kinematics of a Point The value of s H is given by Eq (2. 44)... cos yjX 2X18 63 Dynamics 2 Kinematics of a Point Figure 2. 10 Figure 2. 11 64 Dynamics Dynamics If the path in the x y plane is described by a function y y x , it can be shown that the instantaneous radius of curvature is given by 4 2 53a2 dy 1 dx 2 X 2X19 r d y dx 2 2. 6 .2 CIRCULAR MOTION The point P moves in a plane circular path of radius R as shown in Fig 2. 12 The distance...57 2 Kinematics of a Point rotates is Dy y t Dt À y t The triangle in Fig 2. 3a is isosceles, so the magnitude of Du is jDuj 2juj sin Dya2 2 sin Dya2X The vector Du is where n is a unit vector that points in the direction of Du (Fig 2. 3a) The time derivative of u is du Du 2 sin Dya2n sin Dya2 Dy lim lim lim n Dt 30 Dt Dt 30 Dt 30 dt Dt Dya2 Dt sin Dya2 Dy Dy dy n ... RyY 2X20 Figure 2. 12 where the angle y speci®es the position of the point P along the circular path The velocity is obtained taking the time derivative of Eq (2. 20), v s R y RoY 2X21 where o y is the angular velocity of the line from the center of the path O to the point P The tangential component of the acceleration is at d vadt , and at v R o RaY 2X 22 65 2 Kinematics... obtained differentiating Eq (2. 44): rH Á rHH rH Á rHH X 1a2 sH rH Á rH 2X48 The expression for the normal vector is obtained by substituting Eq (2. 48) into Eq (2. 47): r rHH s H 2 À rH rH Á rHH X 2X49 n s H 4 Because n Á n 1, the radius of curvature is 1 r jrHH s H 2 À rH rH Á rHH j r s H 4 r rHH Á rHH s H 4 À 2 rH Á rHH 2 s H 2 rH rH Á rHH 1a2 Y s H 4 which simpli®es... radial and transverse components, Newton's second law may be written in the form Fr ur Fy uy m ar ur ay uy Y 3X 12 where 2 d 2r dy r À r o2 ar 2 À r dt dt d 2y dr d y r a 2 oX r ay r 2 2 dt dt dt Two scalar equations are obtained: r Fr m À r o2 Fy m r a 2r oX 3X13 The sum of the forces in the radial direction equals the product of the mass and the radial component . dy dx 2 45 3a2 d 2 y dx 2 X 2X19 2. 6 .2 CIRCULAR MOTION The point P moves in a plane circular path of radius R as shown in Fig. 2. 12. The distance s is s Ry Y 2X20 where. Time 52 1 .2 Numbers 52 1.3 Angular Units 53 2. Kinematics of a Point 54 2. 1 Position, Velocity, and Acceleration of a Point 54 2. 2 Angular Motion of a Line 55 2. 3 Rotating Unit Vector 56 2. 4 Straight. dvadt, and a t v R o RaY 2X 22 Figure 2. 12 64 Dynamics Dynamics where a o is the angular acceleration. The normal component of the acceleration is a n v 2 R Ro 2 X 2X23 For the circular