Engineering Mechanics: Dynamics Engineering Mechanics: DynamicsWork of a Force Method of work and energy:directly relates force, mass, velocity and displacement.13 - 1 Work of the force is dz F dy F dx Fds Fr d F dUz y x+ + == = cosrrEngineering Mechanics: Dynamics Engineering Mechanics: DynamicsWork of a Force Work of the force of gravity,dy Wdz F dy F dx F dUyz y x =+ + =13 - 2( ) y W y y Wdy W Uyy = = = 1 22 121 Work of the weight is positive when y < 0, i.e., when the weight moves down.Engineering Mechanics: Dynamics Engineering Mechanics: DynamicsWork of a Force Work of the force exerted by spring,2 22dx kx dx F dUx = =13 - 3222121212 121kx kx dx kx Uxx = = Work of the force exerted by spring is positive when x2 < x1, i.e., when the spring is returning to its undeformed position.Engineering Mechanics: Dynamics Engineering Mechanics: DynamicsParticle Kinetic Energy: Principle of Work & Energydv mv ds Fdsdvmvdtdsdsdvmdtdvm ma Ft t== == = Consider a particle of mass m acted upon by force. The componentFn does no work.Fr13 - 4dv mv ds Ft= Integrating from A1to A2,energy kinetic mv T T T Umv mv dv v m ds Fvvsst= = = = = 2211 2 2 1212122212121 The work of the forceis equal to the change in kinetic energy of the particle.FrEngineering Mechanics: Dynamics Engineering Mechanics: DynamicsPower and Efficiency rate at which work is done.v Fdtr d FdtdUPowerrrrr == == Dimensions of power are work/time or force*velocity.Units for power are13 - 5W 746slb ft550 hp 1 orsmN 1sJ1 (watt) W1 == = =input power output power input workk output worefficiency=== Engineering Mechanics: Dynamics Engineering Mechanics: DynamicsPotential Energy2 1 2 1y W y W U = Work of the force of gravity,Wr Work is independent of path followed; depends only on the initial and final values of Wy.==Wy Vgpotential energy of the body with respect 13 - 6= potential energy of the body with respect to force of gravity.( ) ( )2 12 1 g gV V U = Choice of datum from which the elevation y is measured is arbitrary.Engineering Mechanics: Dynamics Engineering Mechanics: DynamicsPotential Energy Work of the force exerted by a spring depends only on the initial and final deflections of the spring,222121212 1kx kx U = The potential energy of the body with respect to the elastic force,13 - 7to the elastic force,( ) ( )2 12 1221e eeV V Ukx V ==Engineering Mechanics: Dynamics Engineering Mechanics: DynamicsConservative Forces Concept of potential energy can be applied if the work of the force is independent of the path followed by its point of application. ( ) ( )2 2 2 1 1 1 2 1, , , , z y x V z y x V U =Such forces are described as conservative forces. For any conservative force applied on a closed path,13 - 8 For any conservative force applied on a closed path,0 = r d FrrEngineering Mechanics: Dynamics Engineering Mechanics: DynamicsConservation of Energy Work of a conservative force, 2 1 2 1V V U = Concept of work and energy,1 2 2 1T T U = Follows that2 2 1 1+ = + V T V T13 - 9constant2 2 1 1= + =+ = +V T EV T V T When a particle moves under the action of conservative forces, the total mechanical energy is constant. Friction forces are not conservative.Total mechanical energy of a system involving friction decreases.Engineering Mechanics: Dynamics Engineering Mechanics: DynamicsPrinciple of Impulse and Momentum From Newtons second law,( ) = = v m v mdtdFr rrlinear momentum( ) v m d dt Frr= Method of impulse and momentum:directly relates force, mass, velocity, and time.13 - 102 2 1 12 1force the of impulse21v m v mF dt Fttr rr r= += = ImpImp The final momentum of the particle can be obtained by adding vectorially its initial momentum and the impulse of the force during the time interval.1 221v m v m dt Fttr rr = Nonimpulsive forces are forces for whichis small and therefore, may be neglected. For example, the impulse of the gravity force on the ballt FrEngineering Mechanics: Dynamics Engineering Mechanics: DynamicsImpulsive 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,2 1v m t F v mrrr= +13 - 11 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 whichis small and therefore, may be neglected.t FrEngineering Mechanics: Dynamics Engineering Mechanics: DynamicsDirect Central Impact Bodies moving in the same straight line, vA >vB . Upon impact the bodies are in contact and are moving at a common velocity. The bodies either regain their original shape or remain permanently deformed.13 - 12 Wish to determine the final velocities of the two bodies. The total momentum of the two body system is preserved,B B A A B B A Av m v m v m v m+= +Engineering Mechanics: Dynamics Engineering Mechanics: DynamicsDirect Central Impact The second relation between the final velocities.e = Coefficient of Restitution( )B A A Bv v e v v = A second relation between the final velocities is required.13 - 13e = Coefficient of Restitution0 e 1 Perfectly plastic impact, e = 0: v v vA B== ( )v m m v m v mB A B B A A+ = + Perfectly elastic impact, e = 1:Total energy and total are momentum conserved.B A A Bv v v v =