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CHAPTER 13 Kinetics of Particles: Energy and Momentum Methods 13.1 INTRODUCTION Method of work and energy – Involves relations between displacement, velocity, mass, and force. Method of impulse and momentum – Involves relations between mass, velocity, force, and time. Some problems become easier to do. 13.2 WORK OF A FORCE When a force is applied to a mass and the mass moves through an incremental distance, the work done by the force is dU F dr Fx dx Fy dy Fz dz dU F dr F (cos )ds F A r dr A’ r dr dU 0 if 900 dU 0 if 900 dU 0 if 900 Units ft lb or N m Joule ( J ) To get the total work done along a path requires F dr 2 U 12 1 Notice that ds dr 2 2 1 1 U 12 F (cos )ds Ft ds When using rectangular coordinates F 2 U 12 ( Fx dx Fy dy Fz dz ) 1 Ft 0 s1 s2 s The work is the area under the curve. Work of a Constant Force in Rectilinear Motion U 12 F (cos )x F(cos )( x2 x1 ) F 1 x 2 Work of the Force of Gravity dU Wˆj ( dxî dyˆj dzk̂ ) dU Wdy 2 U 12 Wdy W ( y1 y2 ) 1 A2 W U 12 Wy dy A y2 A1 y y1 Work of the Force Exerted by a Spring Spring undeformed x1 F 2 U 12 kxdx ( 21 kx22 21 kx12 ) x 1 x2 12 k ( x2 x1 )( x2 x1 ) F x x1 F1 F2 F kx F kx x2 x Work of a Gravitational Force 2 U 12 GMm GMm GMm 2 dr r2 r1 r 1 Forces which do not do work (ds = 0 or cos a = 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. 13.3 KINETIC ENERGY OF A PARTICLE. PRINCIPLE OF WORK AND ENERGY If the force doing work is the net force then dv dv ds dv mv Ft mat m m dt ds dt ds 2 U 12 2 2 dv Ft ds mv ds mvdv ds 1 1 1 U 12 mv mv 1 2 T 21 mv2 2 2 1 2 U 12 T2 T1 2 1 T1 U 12 T2 13.4 APPLICATIONS OF THE PRINCIPLE OF WORK AND ENERGY Work the pendulum problem in the text. • Wish to determine velocity of pendulum bob at A2. Consider work & kinetic energy. T1 U12 T2 0 Wl 1W 2 v2 2 g v2 2 gl • 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. • 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 A2 , Fn m an v2 2 gl W v22 P W g l W 2 gl P W 3W g l 13.5 POWER AND EFFICIENCY Power is the rate at which work is done. dU F dr P F v dt dt Units – 1 hp = 550 ft lb/s and 1 Watt = 1 J/s Efficiency work out power out work in power in 13.6 POTENTIAL ENERGY Close to the Earth 2 U 12 Wdy W ( y1 y2 ) 1 Define Vg Wy Then U12 (Vg )1 (Vg ) 2 U 12 Vg Not So Close to the Earth 2 U 12 GMm GMm GMm 2 dr r2 r1 r 1 Define 2 GMm WR Vg r r Then U12 (Vg )1 (Vg ) 2 U 12 Vg For a Spring 2 U 12 kxdx ( 21 kx22 21 kx12 ) 1 Define Ve kx 1 2 2 Then U12 (Ve )1 (Ve ) 2 U 12 Ve Notice that the work done by each of these three forces is equal to a change in something that is a function of position only. The idea of a function of position is valid as long as F is conservative. It would not work for a force like friction. This function, V, is call the potential energy. Potential energy is an energy of position or configuration. 13.7 CONSERVATIVE FORCES Work done by conservative forces is independent of the path over which work is done. U12 V ( x1 , y1 , z1 ) V ( x2 , y2 , z2 ) For short U12 V1 V2 For any conservative force F dr 0 An elemental work corresponding to an elemental displacement dU V ( x , y , z ) V ( x dx , y dy , z dz ) dU dV ( x, y, z ) dU dV ( x, y, z ) V V V Fx dx Fy dy Fz dz dx dy dz y z x V Fx x V Fy y V Fz z F Fx î Fy ˆj Fz k̂ V î V ˆj V y z x k̂ The vector in the parentheses is known as the gradient of the scalar function V. F grad V F V 13.8 CONSERVATION OF ENERGY Kinetic plus Potential If the only forces doing work on a system of particles are conservative, then the total mechanical energy is conserved. U 12 V T V1 V2 T2 T1 T1 V1 T2 V2 Pendulum Motion l T1 V1 T2 V2 0 Wl mv 0 1 2 v 2 gl 2 13.9 MOTION UNDER A CONSERVATIVE CENTRAL FORCE. APPLICATION TO SPACE MECHANICS The gravitational attractive force is conservative. So, in space mechanics both energy and angular momentum are conserved since this force is both conservative and central. v P r O r0 mv0 sin 0 rmv sin v0 T0 V0 T V r0 0 P0 1 2 2 GMm 1 mv02 GMm mv 2 r r 0 13.10 PRINCIPLE OF IMPULSE AND MOMENTUM d F ( mv ) dt t2 mv2 mv1 Fdt Imp 12 t1 t2 mv1 Fdt mv2 t1 For a system of particles external impulses are considered only (remember Newton’s third law) If no external forces act on the particles then mv2 mv1 13.11 IMPULSE MOTION Non-impulsive forces can be neglected for they are small in comparison (usually) to the impulsive forces. If it is not known for sure that the forces are small, then include them. Impulsive Motion • Force acting on a particle during a very short time interval that is large enough to cause a significant change in momentum is called an impulsive force. • When impulsive forces act on a particle, mv1 Ft mv2 • When a baseball is struck by a bat, contact occurs over a short time interval but force is large enough to change sense of ball motion. • Nonimpulsive forces are forces for which F t is small and therefore, may be neglected. 13.12 IMPACT • Impact: Collision between two bodies which occurs during a small time interval and during which the bodies exert large forces on each other. • Line of Impact: Common normal to the surfaces in contact during impact. Direct Central Impact • Central Impact: Impact for which the mass centers of the two bodies lie on the line of impact; otherwise, it is an eccentric impact. • Direct Impact: Impact for which the velocities of the two bodies are directed along the line of impact. Oblique Central Impact • Oblique Impact: Impact for which one or both of the bodies move along a line other than the line of impact. • Block constrained to move along horizontal surface. • Impulses from internal forces F and F along the n axis and from external force Fext exerted by horizontal surface and directed along the vertical to the surface. • Final velocity of ball unknown in direction and magnitude and unknown final block velocity magnitude. Three equations required. 13.13 DIRECT CENTRAL IMPACT • Bodies moving in the same straight line, vA > vB . • Upon impact the bodies undergo a period of deformation, at the end of which, they are in contact and moving at a common velocity. • A period of restitution follows during which the bodies either regain their original shape or remain permanently deformed. vA vA • Wish to determine the final velocities of the two bodies. The total momentum of the two body system is preserved, mAvA mB vB mAvA mB vB • A second relation between the final velocities is required. e coefficient of restitution • Period of deformation: • Period of restitution: mAv A Pdt mAu mAu Rdt mAvA • A similar analysis of particle B yields • Combining the relations leads to the desired second relation between the final velocities. • Perfectly plastic impact, e = 0: • Perfectly elastic impact, e = 1: Total energy and total momentum conserved. Rdt e Pdt e u vA vA u vB u u vB vB u u vA vB vA e u vB v A u v A vB vB vA e v A vB vB vA evA vB 0 e1 vB vA v mAvA mB vB mA mB v v v v v B A A B 13.14 OBLIQUE CENTRAL IMPACT • No tangential impulse component; tangential component of momentum for each particle is conserved. • Normal component of total momentum of the two particles is conserved. • Normal components of relative velocities before and after impact are related by the coefficient of restitution. • Final velocities are unknown in magnitude and direction. Four equations are required. v v A t A t v v B t B t mA vA n mB vB n mA vA n mB vB n v v B n A n evA n vB n v v • Tangential momentum of ball is conserved. • Total horizontal momentum of block and ball is conserved. • Normal component of relative velocities of block and ball are related by coefficient of restitution. B B t t mA vA mB vB x mA vA mB vB x v v B n A n evA n vB n • Note: Validity of last expression does not follow from previous relation for the coefficient of restitution. A similar but separate derivation is required. 13.15 PROBLEMS INVOLVING ENERGY AND MOMENTUM • Three methods for the analysis of kinetics problems: - Direct application of Newton’s second law - Method of work and energy - Method of impulse and momentum • Select the method best suited for the problem or part of a problem under consideration.