5/2/2015 7:42 AM. Chapter 17. Plane Motion of Rigid Bodies: Energy and Momentum Methods. Mohammad Suliman Abuhaiba, Ph.D., PE

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1 5//05 7:4 AM Chapter 7 Plane Motion of Rigid Bodies: Energy and Momentum Methods

2 5//05 7:4 AM Chapter Outline Principle of Work and Energy for a Rigid Body Work of Forces Acting on a Rigid Body Kinetic Energy of a Rigid Body in Plane Motion Conservation of Energy Power Principle of Impulse and Momentum Conservation of Angular Momentum Eccentric Impact

3 3 5//05 7:4 AM Introduction Principle of work & energy: solution of problems involving displacements & velocities T U T Principle of impulse & momentum: velocities & time L t t Fdt L t H O MOdt H O Problems involving eccentric impact:. Principle of impulse and momentum. Application of coefficient of restitution t

4 4 5//05 7:4 AM Principle of Work & Energy for a RB RB is made of a large number of particles. T U T T, T U initial & final total kinetic energy of particles forming body total work of internal & external forces acting on particles of body

5 5 5//05 7:4 AM Principle of Work & Energy for a RB Internal forces between particles A & B are equal and opposite. Small displacements of particles A & B are not equal but components of displacements along AB are equal. net work of internal forces is zero

6 6 5//05 7:4 AM Principle of Work & Energy for a RB Two forces F and F forming a couple of moment during a displacement of their points of application. du F dr F dr F dr F ds M Fr d M d U M d M if M is constant.

7 7 5//05 7:4 AM Work of Forces Acting on a RB Forces acting on rigid bodies which do no work: Forces applied to fixed points: reactions at a frictionless pin when the supported body rotates about the pin. Forces acting in a direction perpendicular to displacement of their point of application: reaction at a frictionless surface to a body moving along the surface weight of a body when its CG moves horizontally

8 8 5//05 7:4 AM Work of Forces Acting on a RB Forces acting on rigid bodies which do no work: Friction force at point of contact of a body rolling without sliding on a fixed surface. du F ds C F v dt 0 c

9 9 5//05 7:4 AM T Kinetic Energy of a RB in Plane Motion mv mv mv I Δm v r i i i Δm i Kinetic energy of a RB:. kinetic energy associated with motion of mass center G. kinetic energy associated with rotation of body about G.

10 Kinetic Energy of a RB in Plane Motion 5//05 7:4 AM 0 A RB rotating about a fixed axis through O Δ Δ Δ O i i i i i i I m r r m m v T

11 5//05 7:4 AM Systems of Rigid Bodies Principle of work and energy can be applied to each body. Principle of work and energy may also applied to the entire system, T,T U T U T = sum of kinetic energies of all bodies forming the system = work of all forces acting on various bodies (internal or external)

12 5//05 7:4 AM Systems of Rigid Bodies For problems involving pin connected members, blocks and pulleys connected by inextensible cords, and meshed gears, internal forces occur in pairs of equal and opposite forces points of application of each pair move through equal distances net work of internal forces is zero work on system reduces to work of external forces

13 Conservation of Energy T V 3 T V 5//05 7:4 AM T 0, V 0 T mass m released with zero velocity determine at mv ml m l ml 3 V sin Wl mgl sin T V I T V ml 0 3 3g sin l mgl sin

14 4 5//05 7:4 AM Power Power = rate at which work is done For a body acted upon by force and moving with velocity, du Power F v dt For a RB rotating with an angular velocity and acted upon by a couple of moment parallel to axis of rotation, du M d Power dt v dt M F M

15 5 5//05 7:4 AM Sample Problem 7. For the drum & flywheel, I = 0.5 lb.ft.s. The bearing friction is equivalent to a couple of 60 lb.ft. At the instant shown, the block is moving downward at 6 ft/s. Determine velocity of block after it has moved 4 ft downward.

16 6 5//05 7:4 AM Sample Problem 7. The system is at rest when a moment M = 6 N.m is applied to gear B. Neglecting friction, determine a. number of revolutions of gear B before its angular velocity reaches 600 rpm b. tangential force exerted by gear B on gear A. m m A B 0kg 3kg k k A B 00mm 80mm

17 7 5//05 7:4 AM Sample Problem 7.3 A sphere, cylinder, and hoop, each having the same mass and radius, are released from rest on an incline. Determine the velocity of each body after it has rolled through a distance corresponding to a change of elevation h.

18 8 5//05 7:4 AM Sample Problem 7.4 A 30-lb slender rod pivots about point O. The other end is pressed against a spring (k = 800 lb/in) until spring is compressed one inch and rod is in a horizontal position. If rod is released from this position, determine its angular velocity and reaction at the pivot as the rod passes through a vertical position.

19 9 5//05 7:4 AM Sample Problem 7.5 Each of the two slender rods has a mass of 6 kg. The system is released from rest with b = 60 o. Determine a. angular velocity of rod AB when b = 0 o b. velocity of point D at the same instant.

20 0 5//05 7:4 AM Assignment 7-3,, 8, 5, 3, 40, 47

21 5//05 7:4 AM Principle of Impulse and Momentum Solution of problems involving time and velocity Problems involving impulsive motion and impact Sys Momenta + Sys Ext Imp - = Sys Momenta

22 Principle of Impulse and Momentum 5//05 7:4 AM Momenta of particles of a system may be reduced to a vector attached to the mass center equal to their sum, L v Δm i i mv and a couple equal to sum of their moments about the mass center, HG r i v iδmi For plane motion of a rigid slab or of a rigid body symmetrical with respect to the reference plane, H G I

23 3 5//05 7:4 AM Principle of Impulse and Momentum Plane motion of a rigid slab, 3 equations of motion: momenta and impulses in x and y directions moments of momenta and impulses wrt any given point

24 4 5//05 7:4 AM Principle of Impulse and Momentum Non-centroidal rotation: Equating moments of momenta and impulses about O, I O t M dt IO t O

25 5 5//05 7:4 AM Conservation of Angular Momentum When no external force acts on a RB or a system of rigid bodies, the system of momenta at t is equipollent to the system at t. Total linear momentum and angular momentum about any point are conserved, L L H 0 H0 When sum of angular impulses pass through O, linear momentum may not be conserved, yet angular momentum about O is conserved, H 0 H0

26 6 5//05 7:4 AM Sample Problem 7.6 The system is at rest when a moment of M = 6 N.m is applied to gear B. Neglecting friction, a. determine time required for gear B to reach an angular velocity of 600 rpm b. tangential force exerted by gear B on gear A. m m A B 0kg 3kg k k A B 00mm 80mm

7 5//05 7:4 AM Sample Problem 7.7 Uniform sphere of mass m and radius r is projected along a rough horizontal surface with a linear velocity and no angular velocity.

27 7 5//05 7:4 AM Sample Problem 7.7 Uniform sphere of mass m and radius r is projected along a rough horizontal surface with a linear velocity and no angular velocity. The coefficient of kinetic friction is m k. Determine v a. time t at which the sphere will start rolling without sliding b. linear and angular velocities of the sphere at time t.

28 8 5//05 7:4 AM Sample Problem 7.8 Two solid spheres (radius = 3in, W = -lb) are mounted on a spinning horizontal rod ( = 6 rad/s). The balls are held together by a string which is suddenly cut. Determine a. angular velocity of rod after the balls have moved to A and B b. energy lost due to the plastic impact of the spheres and stops. I R 0.5 lb ft s

29 9 5//05 7:4 AM Assignment 7-54, 60, 69, 76, 83, 9

30 Eccentric Impact 30 5//05 7:4 AM u A n ub n Period of restitution Period of deformation Impulse Impulse Rdt Pdt Rdt vb v n A e coefficien t of restitutio n Pdt v v n A n B n

31 3 5//05 7:4 AM Sample Problem 7.9 A 0.05-lb bullet is fired into the side of a 0-lb square panel which is initially at rest. Determine a. angular velocity of panel immediately after bullet becomes embedded b. impulsive reaction at A, assuming that bullet becomes embedded in s.

32 3 5//05 7:4 AM Sample Problem 7.0 A -kg sphere with an initial velocity of 5 m/s strikes the lower end of an 8-kg rod AB. The rod is hinged at A and initially at rest. The coefficient of restitution between the rod and sphere is 0.8. Determine the angular velocity of the rod and the velocity of the sphere immediately after impact.

33 33 5//05 7:4 AM Sample Problem 7. A square package of mass m moves down conveyor belt A with constant velocity. At end of the conveyor, the corner of the package strikes a rigid support at B. The impact is perfectly plastic. Derive an expression for the minimum velocity of conveyor belt A for which the package will rotate about B and reach conveyor belt C.

34 34 5//05 7:4 AM Assignment , 04,,, 30

DYNAMICS VECTOR MECHANICS FOR ENGINEERS: Plane Motion of Rigid Bodies: Energy and Momentum Methods. Seventh Edition CHAPTER

DYNAMICS VECTOR MECHANICS FOR ENGINEERS: Plane Motion of Rigid Bodies: Energy and Momentum Methods. Seventh Edition CHAPTER CHAPTER 7 VECTOR MECHANICS FOR ENGINEERS: DYNAMICS Ferdinand P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Texas Tech University Plane Motion of Rigid Bodies: Energy and Momentum Methods