ME 141. Lecture 11: Kinetics of particles: Energy method

Transcription

1 ME 4 Engineering Mechanics Lecture : Kinetics of particles: Energy method Ahmad Shahedi Shakil Lecturer, Dept. of Mechanical Engg, BUE sshakil@me.buet.ac.bd, shakil679@gmail.com ebsite: teacher.buet.ac.bd/sshakil Courtesy: Vector Mechanics for Engineers, Beer and Johnston

2 Introduction Approaches to Kinetics Problems Forces and Accelerations Velocities and Displacements Velocities and ime Newton s Second Law (last chapter) ork-energy Impulse- Momentum F ma G U t mv F dt mv t

3 ork of a Force Differential vector dr is the particle displacement. ork of the force is du F dr F ds cos F x dx F y dy F z dz ork is a scalar quantity, i.e., it has magnitude and sign but not direction. Dimensions of work are length force. Units are joule N m ft lb.356 J J

4 ork of a Force ork of a force during a finite displacement, U A A s s A Fxdx Fydy Fzdz A F dr F cos ds s s F t ds ork is represented by the area under the curve of F t plotted against s. F t is the force in the direction of the displacement ds

5 ork of a Force ork of the force of gravity, U du F x dy y y dx F dy y dy F dz y y y z ork of the weight is equal to product of weight and vertical displacement y. In the figure above, when is the work done by the weight positive? a) Moving from y to y b) Moving from y to y c) Never

6 ork 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. ork of the force exerted by spring, du U F dx kx dx x x kx dx ork of the force exerted by spring is positive when x < x, i.e., when the spring is returning to its undeformed position. kx kx ork of the force exerted by the spring is equal to negative of area under curve of F plotted against x, U F F x

7 ork of a Force Forces which do not do work (ds = 0 or cos 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 eight of a body when its center of gravity moves horizontally.

8 Particle Kinetic Energy: Principle of ork & Energy Consider a particle of mass m acted upon by force F dv F ma m F t t ds t dv m ds mv dv ds dt dt dv mv ds Integrating from A to A, s s F ds t m v v v dv mv mv U mv kinetic energy he work of the force F is equal to the change in kinetic energy of the particle. Units of work and kinetic energy are the same: m m mv kg kg m N m J s s

9 Power and Efficiency Power rate at which work is done. du F dr dt dt F v Dimensions of power are work/time or force*velocity. Units for power are J m ft lb (watt) N or hp 550 s s s efficiency output work input work power output power input 746

10 Sample Problem 3. SOLUION: Evaluate the change in kinetic energy. Determine the distance required for the work to equal the kinetic energy change. An automobile weighing 4000 lb is driven down a 5 o incline at a speed of 60 mi/h when the brakes are applied causing a constant total breaking force of 500 lb. Determine the distance traveled by the automobile as it comes to a stop.

11 Sample Problem 3. SOLUION: Evaluate the change in kinetic energy. v v 60 U mv mi h 580 ft mi ft lb h 3600 s 88ft ft lb Determine the distance required for the work to equal the kinetic energy change. U 0 500lbx 4000lbsin 5 5lbx 0 5lbx 0 x s x 48 ft

12 Sample Problem 3. wo blocks are joined by an inextensible cable as shown. If the system is released from rest, determine the velocity of block A after it has moved m. Assume that the coefficient of friction between block A and the plane is m k = 0.5 and that the pulley is weightless and frictionless. SOLUION: Apply the principle of work and energy separately to blocks A and B. hen the two relations are combined, the work of the cable forces cancel. Solve for the velocity.

13 Sample Problem 3. SOLUION: Apply the principle of work and energy separately to blocks A and B. F A A U 0 F B F 00kg 9.8m s m N C U 0 F F k C A m : k m F m A A 96 N m 490 Nm 00kgv 300 kg 9.8m s c c : m m B N m A v 490 N m 940 N m 300 kgv 940 N m B v

14 Sample Problem 3. hen the two relations are combined, the work of the cable forces cancel. Solve for the velocity. F C F c m 490 Nm 00kgv m 940 Nm 300kgv 940 Nm 490 Nm 00kg 300kg 4900 J 500kgv v v 4.43 m s

15 Sample Problem 3.3 A spring is used to stop a 60 kg package which is sliding on a horizontal surface. he spring has a constant k = 0 kn/m and is held by cables so that it is initially compressed 0 mm. he package has a velocity of.5 m/s in the position shown and the maximum deflection of the spring is 40 mm. Determine (a) the coefficient of kinetic friction between the package and surface and (b) the velocity of the package as it passes again through the position shown. SOLUION: Apply the principle of work and energy between the initial position and the point at which the spring is fully compressed and the velocity is zero. he only unknown in the relation is the friction coefficient. Apply the principle of work and energy for the rebound of the package. he only unknown in the relation is the velocity at the final position.

16 Sample Problem 3.3 SOLUION: Apply principle of work and energy between initial position and the point at which spring is fully compressed. kg.5m s 87.5J 0 mv 60 U f m m k x 60 kg 9.8m s m 377 J k k Pmin kx0 0 kn m0.0 m Pmax kx0 x 0 kn m0.60 m U P P x U e min max 400 N 300 N 400 N 300 N0.040 m.0j U U 377 Jm J f e k U 87.5J - : 377 J J 0 m 0. 0 k m k m

17 Sample Problem 3.3 Apply the principle of work and energy for the rebound of the package mv 60kg v U 3 3 U U 377 J J f 3 e m k J U J 3 : 60 kg v 3 v 3.03m s

18 Sample Problem 3.4 SOLUION: A 000 lb car starts from rest at point and moves without friction down the track shown. Determine: a) the force exerted by the track on the car at point, and b) the minimum safe value of the radius of curvature at point 3. Apply principle of work and energy to determine velocity at point. Apply Newton s second law to find normal force by the track at point. Apply principle of work and energy to determine velocity at point 3. Apply Newton s second law to find minimum radius of curvature at point 3 such that a positive normal force is exerted by the track.

19 Sample Problem 3.4 SOLUION: Apply principle of work and energy to determine velocity at point. s 50.8ft 3.ft 40 ft 40 ft 40 ft 0 : 40 ft 0 v s g v v g U U v g mv Apply Newton s second law to find normal force by the track at point. : n a n F m N g g v g ma N n 5 0ft 40ft N 0000 lb

20 Sample Problem 3.4 Apply principle of work and energy to determine velocity at point 3. v 3 U ft 5ftg 5ft3.ft s v 40.ft s v g Apply Newton s second law to find minimum radius of curvature at point 3 such that a positive normal force is exerted by the track. 3 3 Fn ma n : ma g n 3 v3 5ft g 3 g 3 50ft

21 Sample Problem 3.5 SOLUION: Force exerted by the motor cable has same direction as the dumbwaiter velocity. Power delivered by motor is equal to Fv D, v D = 8 ft/s. he dumbwaiter D and its load have a combined weight of 600 lb, while the counterweight C weighs 800 lb. Determine the power delivered by the electric motor M when the dumbwaiter (a) is moving up at a constant speed of 8 ft/s and (b) has an instantaneous velocity of 8 ft/s and an acceleration of.5 ft/s, both directed upwards. In the first case, bodies are in uniform motion. Determine force exerted by motor cable from conditions for static equilibrium. In the second case, both bodies are accelerating. Apply Newton s second law to each body to determine the required motor cable force.

22 Sample Problem 3.5 In the first case, bodies are in uniform motion. Determine force exerted by motor cable from conditions for static equilibrium. Free-body C: F y 0 : 800lb lb Free-body D: F y 0 : F 600 lb F 600 lb lb 400 lb 00 lb Power Fv D 600 ft lb 00 lb8ft s s Power hp 550ft lb 600 ft lb s.9hp s

23 Sample Problem 3.5 In the second case, both bodies are accelerating. Apply Newton s second law to each body to determine the required motor cable force. a D.5ft s ac ad.5ft Free-body C: Fy mc ac : lb Free-body D: 600 Fy mdad : F F s F 6.lb Power Power FvD 6.lb8ft s 097 ft lb s hp 550ft lb 097ft lb s 3.8hp s

24 Prob # 3. Packages are thrown down an incline at A with a velocity of m/s. he packages slide along the surface ABC to a conveyor belt which moves with a velocity of m/s. Knowing that μ k = 0.5 between the packages and the surface ABC, determine the distance d if the packages are to arrive at C with a velocity of m/s

25 Prob # 3.0 he system shown is at rest when a constant 30-lb force is applied to collar B. (a) If the force acts through the entire motion, determine the speed of collar B as it strikes the support at C. (b) After what distance d should the 30-lb force be removed if the collar is to reach support C with zero velocity?

26 Potential Energy ork of the force of gravity, U y y ork is independent of path followed; depends only on the initial and final values of y. y V g U potential energy of the body with respect to force of gravity. V g V g Choice of datum from which the elevation y is measured is arbitrary. Units of work and potential energy are the same: y N m J V g

27 Potential Energy ork of the force exerted by a spring depends only on the initial and final deflections of the spring, U kx kx he potential energy of the body with respect to the elastic force, V U e kx Ve Ve Note that the preceding expression for V e is valid only if the deflection of the spring is measured from its undeformed position.

28 Conservative 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. U x, y, z V x, y z V, Such forces are described as conservative forces. For any conservative force applied on a closed path, F dr 0 Elementary work corresponding to displacement between two neighboring points, du F x F V dx F V x x, y, zv x dx, y dy, z dz dvx, y, y dy F V y z z V dz x V z gradv V dx y V dy z dz

29 Conservation of Energy ork of a conservative force, V V U Concept of work and energy, U Follows that constant V E V V hen a particle moves under the action of conservative forces, the total mechanical energy is constant. V V 0 V V g g mv 0 Friction forces are not conservative. otal mechanical energy of a system involving friction decreases. Mechanical energy is dissipated by friction into thermal energy. otal energy is constant.

30 Sample Problem 3.6 SOLUION: Apply the principle of conservation of energy between positions and. he elastic and gravitational potential energies at and are evaluated from the given information. he initial kinetic energy is zero. A 0 lb collar slides without friction along a vertical rod as shown. he spring attached to the collar has an undeflected length of 4 in. and a constant of 3 lb/in. Solve for the kinetic energy and velocity at. If the collar is released from rest at position, determine its velocity after it has moved 6 in. to position.

31 Sample Problem 3.6 SOLUION: Apply the principle of conservation of energy between positions and. Position : V kx 3lb in. 8 in. 4 in. V e V 0 e V g 4in. lb 0 ft lb Position : V e kx 3lb in. 0 in. 4 in. V y 0 lb 6 in. 0 in. V g V e V mv g Conservation of Energy: V V ft lb 0.3v 0.3v 4in. lb in. lb 5.5 ft lb v 5.5 ft lb v 54in. lb lb 4.9ft s

32 Prob# 3.64 A -kg collar is attached to a spring and slides without friction in a vertical plane along the curved rod ABC. he spring is undeformed when the collar is at C and its constant is 600 N/m. If the collar is released at A with no initial velocity, determine its velocity (a) as it passes through B, (b) as it reaches C.

33 Prob# 3.70 A section of track for a roller coaster consists of two circular arcs AB and CD joined by a straight portion BC. he radius of AB is 7 m and the radius of CD is 7 m. he car and its occupants, of total mass 50 kg, reach point A with practically no velocity and then drop freely along the track. Determine the normal force exerted by the track on the car as the car reaches point B. Ignore air resistance and rolling resistance.

34 Prob # 3.68 A spring is used to stop a 50-kg package which is moving down a 0 incline. he spring has a constant k = 30 kn/m and is held by cables so that it is initially compressed 50 mm. Knowing that the velocity of the package is m/s when it is 8 m from the spring and neglecting friction, determine the maximum additional deformation of the spring in bringing the package to rest.