Engineering Mechanics: Dynamics - Inside Mines

Engineering Mechanics: DynamicsWork of a Force Method of work and energy: directly relates force, mass, velocityand displacement. Work of the force isr rdU F dr F ds cosα Fxdx Fydy Fz dz13 - 1

Engineering Mechanics: DynamicsWork of a Force Work of the force of gravity,dU Fx dx Fy dy Fz dz W dyy2U1 2 W dyy1 W ( y 2 y1 ) W y Work of the weight is positive when y 0,i.e., when the weight moves down.13 - 2

Engineering Mechanics: DynamicsWork of a Force Work of the force exerted by spring,dU F dx kx dxx2U1 2 kx dx 12 kx12 12 kx22x1 Work of the force exerted by spring is positivewhen x2 x1, i.e., when the spring is returning toits undeformed position.13 - 3

Engineering Mechanics: DynamicsParticle Kinetic Energy: Principle of Work & Energy Consider a particle of mass m acted upon by force.rThe component Fn does no work.FdvFt mat mdtdv dsdv m mvds dtdsF t ds mv dv Integrating from A1 to A2 ,s2v2s1v122 12 mv1 Ft ds m v dv 12 mv2rU1 2 T2 T1FT 12 mv 2 kinetic energy The work of the force is equal to the change inkinetic energy of the particle.13 - 4

Engineering Mechanics: DynamicsPower and Efficiency Power rate at which work is done.r rdU F dr dtdtr r F v Dimensions of power are work/time or force*velocity.Units for power areJm1 W (watt) 1 1 N ssor 1 hp 550ft lb 746 Ws η efficiencyoutput work input workpower output power input13 - 5

Engineering Mechanics: DynamicsPotential Energyr Work of the force of gravity W,U1 2 W y1 W y 2 Work is independent of path followed; dependsonly on the initial and final values of Wy.V g Wy potential energy of the body with respectto force of gravity.( ) ( )U1 2 V g V g12 Choice of datum from which the elevation y ismeasured is arbitrary.13 - 6

Engineering Mechanics: DynamicsPotential Energy Work of the force exerted by a spring dependsonly on the initial and final deflections of thespring,U1 2 12 kx12 12 kx22 The potential energy of the body with respectto the elastic force,Ve 12 kx 2U1 2 (Ve )1 (Ve )213 - 7

Engineering Mechanics: DynamicsConservative Forces Concept of potential energy can be applied if thework of the force is independent of the pathfollowed by its point of application.U1 2 V ( x1 , y1 , z1 ) V ( x2 , y 2 , z 2 )Such forces are described as conservative forces. For any conservative force applied on a closed path,r r F dr 013 - 8

Engineering Mechanics: DynamicsConservation of Energy Work of a conservative force, U1 2 V1 V2 Concept of work and energy,U1 2 T2 T1 Follows thatT1 V1 T2 V2E T V constant When a particle moves under the action of conservative forces, the totalmechanical energy is constant. Friction forces are not conservative. Total mechanical energy of a systeminvolving friction decreases.13 - 9

Engineering Mechanics: DynamicsPrinciple of Impulse and Momentum Method of impulse and momentum: directly relates force, mass, velocity,and time. From Newton’s second law,r drrF (mv )mv linear momentumdtrrFdt d (mv )t2rrr Fdtmvmv 21 Nonimpulsiver forces are forcesfor which F t is small andtherefore, may be neglected. Forexample, the impulse of thegravity force on the ballt1t2rr Fdt Imp1 2 impulse of the force Ft1rrmv1 Imp1 2 mv2 The final momentum of the particle can be obtained by adding vectorially itsinitial momentum and the impulse of the force during the time interval.13 - 10

Engineering Mechanics: DynamicsImpulsive Motion Force acting on a particle during a very shorttime interval that is large enough to cause asignificant change in momentum is called animpulsive force. When impulsive forces act on a particle,rrrmv1 F t mv2 When a baseball is struck by a bat, contactoccurs over a short time interval but force islarge enough to change sense of ball motion. Nonimpulsiveforces are forces for whichrF t is small and therefore, may beneglected.13 - 11

Engineering Mechanics: DynamicsDirect Central Impact Bodies moving in the same straight line,vA vB . Upon impact the bodies are in contact andare moving at a common velocity. The bodies either regain their originalshape or remain permanently deformed. Wish to determine the final velocities of thetwo bodies. The total momentum of the two bodysystem is preserved,m A v A mB v B m A v’A m B v’B13 - 12

Engineering Mechanics: DynamicsDirect Central Impact A second relation between the final velocities is required. The second relation between the finalvelocities.v′B v′A e(v A v B )e Coefficient of Restitution0 e 1 Perfectly plastic impact, e 0: v′B v′A v′m A v A m B v B (m A mB )v′ Perfectly elastic impact, e 1:Total energy and total are momentumconserved.v′B v′A v A v B13 - 13

Engineering Mechanics: Dynamics Work of a Force Work of the force exerted by spring, 2 2 2 dU F dx kx dx x 13 - 3 2 2 1 2 1 1 1 2 1 U kx dx kx kx x Work of the force exerted by spring is positive when x2 x1, i.e.,