Seventh Edition VECTOR MECHANICS FOR ENGINEERS: CHAPTER �� DYNAMICS Ferdinand P. Beer Kinetics of Particles: Energy E. Russell Johnston, Jr. and Momentum Methods Lecture Notes: J. Walt Oler Texas Tech University � ����������������������������������������������������������
Edition Seventh Vector Mechanics for Engineers: Dynamics Contents Introduction Sample Problem 13.6 Work of a Force Sample Problem 13.7 Principle of Work & Energy Sample Problem 13.9 Applications of the Principle of Work & Principle of Impulse and Momentum Energy Impulsive Motion Power and Efficiency Sample Problem 13.10 Sample Problem 13.1 Sample Problem 13.11 Sample Problem 13.2 Sample Problem 13.12 Sample Problem 13.3 Impact Sample Problem 13.4 Direct Central Impact Sample Problem 13.5 Oblique Central Impact Potential Energy Problems Involving Energy and Momentum Conservative Forces Sample Problem 13.14 Conservation of Energy Sample Problem 13.15 Motion Under a Conservative Central Sample Problems 13.16 Force Sample Problem !3.17 � ���������������������������������������������������������� 13 - 2
Edition Seventh Vector Mechanics for Engineers: Dynamics Introduction • Previously, problems dealing with the motion of particles were � � = solved through the fundamental equation of motion, F m a . Current chapter introduces two additional methods of analysis. • Method of work and energy : directly relates force, mass, velocity and displacement. • Method of impulse and momentum : directly relates force, mass, velocity, and time. � ���������������������������������������������������������� 13 - 3
Edition Seventh Vector Mechanics for Engineers: Dynamics Work of a Force d � • Differential vector r is the particle displacement . • Work of the force is � � = • dU F d r = α F ds cos = + + F dx F dy F dz x y z • Work is a scalar quantity, i.e., it has magnitude and sign but not direction. length × force. • Dimensions of work are Units are ( ) ( )( ) = ⋅ = 1 J joule 1 N 1 m 1ft lb 1.356 J � ���������������������������������������������������������� 13 - 4
Edition Seventh Vector Mechanics for Engineers: Dynamics Work of a Force • Work of a force during a finite displacement, A � � 2 � = • U F d r → 1 2 A 1 s s 2 2 ( ) � � = α = F cos ds F ds t s s 1 1 A ( ) 2 � = + + F dx F dy F dz x y z A 1 • Work is represented by the area under the curve of F t plotted against s . � ���������������������������������������������������������� 13 - 5
Edition Seventh Vector Mechanics for Engineers: Dynamics Work of a Force • Work of a constant force in rectilinear motion, ( ) = α ∆ U F cos x → 1 2 • Work of the force of gravity, = + + dU F dx F dy F dz x y z = − W dy y 2 � = − U W dy → 1 2 y 1 ( ) = − − = − ∆ W y y W y 2 1 • Work of the weight is equal to product of weight W and vertical displacement ∆ y. • Work of the weight is positive when ∆ y < 0, i.e., when the weight moves down. � ���������������������������������������������������������� 13 - 6
Edition Seventh Vector Mechanics for Engineers: Dynamics Work of a Force • Magnitude of the force exerted by a spring is proportional to deflection, = F kx ( ) = k spring constant N/m or lb/in. • Work of the force exerted by spring , = − = − dU F dx kx dx x 2 2 2 � = − = − 1 1 U kx dx kx kx → 1 2 1 2 2 2 x 1 • Work of the force exerted by spring is positive when x 2 < x 1 , i.e., when the spring is returning to its undeformed position. • Work of the force exerted by the spring is equal to negative of area under curve of F plotted against x , ( ) = − 1 + ∆ U F F x → 1 2 1 2 2 � ���������������������������������������������������������� 13 - 7
Edition Seventh Vector Mechanics for Engineers: Dynamics Work of a Force Work of a gravitational force (assume particle M occupies fixed position O while particle m follows path shown), Mm = − = − dU Fdr G dr 2 r r 2 Mm Mm Mm � = − = − U G dr G G → 1 2 2 r r r 2 1 r 1 � ���������������������������������������������������������� 13 - 8
Edition Seventh Vector Mechanics for Engineers: Dynamics Work of a Force Forces which do not do work (ds = 0 or cos α = 0) : • reaction at frictionless pin supporting rotating body, • reaction at frictionless surface when body in contact moves along surface, • reaction at a roller moving along its track, and • weight of a body when its center of gravity moves horizontally. � ���������������������������������������������������������� 13 - 9
Edition Seventh Vector Mechanics for Engineers: Dynamics Particle Kinetic Energy: Principle of Work & Energy � • Consider a particle of mass m acted upon by force F dv = = F ma m t t dt dv ds dv = = m mv ds dt ds = F ds mv dv t • Integrating from A 1 to A 2 , s v 2 2 2 2 � � = = 1 − 1 F ds m v dv mv mv t 2 1 2 2 s v 1 1 2 = − = = 1 U T T T mv kinetic energy → 1 2 2 1 2 � • The work of the force is equal to the change in kinetic F energy of the particle . • Units of work and kinetic energy are the same: 2 � � � � m m � � � � = 2 = = = ⋅ = 1 T mv kg kg m N m J � � � � 2 2 s s � ���������������������������������������������������������� 13 - 10
Edition Seventh Vector Mechanics for Engineers: Dynamics Applications of the Principle of Work and Energy • Wish to determine velocity of pendulum bob at A 2 . Consider work & kinetic energy. � • Force acts normal to path and does no work. P + = T U T → 1 1 2 2 1 + = 2 0 ml mv 2 2 = 2 v gl 2 • Velocity found without determining expression for acceleration and integrating. • All quantities are scalars and can be added directly. • Forces which do no work are eliminated from the problem. � ���������������������������������������������������������� 13 - 11
Edition Seventh Vector Mechanics for Engineers: Dynamics Applications of the Principle of Work and Energy • Principle of work and energy cannot be applied to directly determine the acceleration of the pendulum bob. • Calculating the tension in the cord requires supplementing the method of work and energy with an application of Newton’s second law. • As the bob passes through A 2 , � = F m a n n 2 v − = = 2 P mg ma m n l 2 gl = + = P mg m 3 mg 2 = l v 2 gl � ���������������������������������������������������������� 13 - 12
Edition Seventh Vector Mechanics for Engineers: Dynamics Power and Efficiency = • Power rate at which work is done. � � • dU F d r = = dt dt � � = • F v • Dimensions of power are work/time or force*velocity. Units for power are J m = = ⋅ 1 W (watt) 1 1 N s s η = efficiency • output wor k = input work power output = power input � ���������������������������������������������������������� 13 - 13
Edition Seventh Vector Mechanics for Engineers: Dynamics Sample Problem 13.1 SOLUTION: • Evaluate the change in kinetic energy. • Determine the distance required for the work to equal the kinetic energy change. An automobile weighing 19.62 kN is driven down a 5 o incline at a speed of 100 km/h when the brakes are applied causing a constant total breaking force of 7 kN. Determine the distance traveled by the automobile as it comes to a stop. � ���������������������������������������������������������� 13 - 14
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