ME 141. Engineering Mechanics

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of Mechanical Engg, BUET E-mail: hakil@me.buet.ac.bd, hakil6791@gmail.

1 ME 141 Engineering Mechanic Lecture 14: Plane motion of rigid bodie: Force and acceleration Ahmad Shahedi Shakil Lecturer, Dept. of Mechanical Engg, BUET Webite: teacher.buet.ac.bd/hakil Courtey: Vector Mechanic for Engineer, Beer and Johnton

2 Introduction In thi chapter and in Chapter 17 and 18, we will be concerned with the kinetic of rigid bodie, i.e., relation between the force acting on a rigid body, the hape and ma of the body, and the motion produced. Reult of thi chapter will be retricted to: - plane motion of rigid bodie, and - rigid bodie coniting of plane lab or bodie which are ymmetrical with repect to the reference plane. Our approach will be to conider rigid bodie a made of large number of particle and to ue the reult of Chapter 14 for the motion of ytem of particle. Specifically, F ma M G H and G

Equation of Motion for a Rigid Body Conider a rigid body acted upon by everal external force. Aume that the body i made of a large number of particle.

3 Equation of Motion for a Rigid Body Conider a rigid body acted upon by everal external force. Aume that the body i made of a large number of particle. For the motion of the ma center G of the body with repect to the Newtonian frame Oxyz, F ma For the motion of the body with repect to the centroidal frame Gx y z, M G H G Sytem of external force i equipollent to the ytem coniting of ma and H. G

4 Angular Momentum of a Rigid Body in Plane Motion Conider a rigid lab in plane motion. Angular momentum of the lab may be computed by I m r m r r m v r H i i n i i i i n i i i i G Δ Δ Δ 1 1 After differentiation, I I H G Reult are alo valid for plane motion of bodie which are ymmetrical with repect to the reference plane. Reult are not valid for aymmetrical bodie or three-dimenional motion.

Plane Motion of a Rigid Body: D Alembert Principle Motion of a rigid body in plane motion i completely defined by the reultant and moment reultant about G of the external

F x ma x F y ma y M G I The external force and the collective effective force of the lab particle are equipollent (reduce to the ame reultant and moment reultant) and

5 Plane Motion of a Rigid Body: D Alembert Principle Motion of a rigid body in plane motion i completely defined by the reultant and moment reultant about G of the external force. F x ma x F y ma y M G I The external force and the collective effective force of the lab particle are equipollent (reduce to the ame reultant and moment reultant) and equivalent (have the ame effect on the body). d Alembert Principle: The external force acting on a rigid body are equivalent to the effective force of the variou particle forming the body. The mot general motion of a rigid body that i ymmetrical with repect to the reference plane can be replaced by the um of a tranlation and a centroidal rotation.

Axiom of the Mechanic of Rigid Bodie The force F and F act at different point on a rigid body but but have the ame magnitude, direction, and line of action.

6 Axiom of the Mechanic of Rigid Bodie The force F and F act at different point on a rigid body but but have the ame magnitude, direction, and line of action. The force produce the ame moment about any point and are therefore, equipollent external force. Thi prove the principle of tranmiibility wherea it wa previouly tated a an axiom.

Problem Involving the Motion of a Rigid Body The fundamental relation between the force acting on a rigid body in plane motion and the acceleration of it ma

The technique for olving problem of tatic equilibrium may be applied to olve problem of plane motion by utilizing - d Alembert principle, or - principle of

7 Problem Involving the Motion of a Rigid Body The fundamental relation between the force acting on a rigid body in plane motion and the acceleration of it ma center and the angular acceleration of the body i illutrated in a free-body-diagram equation. The technique for olving problem of tatic equilibrium may be applied to olve problem of plane motion by utilizing - d Alembert principle, or - principle of dynamic equilibrium Thee technique may alo be applied to problem involving plane motion of connected rigid bodie by drawing a freebody-diagram equation for each body and olving the correponding equation of motion imultaneouly.

8 Free Body Diagram and Kinetic Diagram Draw the FBD and KD for the bar AB of ma m. A known force P i applied at the bottom of the bar.

9 Free Body Diagram and Kinetic Diagram L/ C G C y A r C x y x I G ma y 1. Iolate body. Axe 3. Applied force 4. Replace upport with force 5. Dimenion 6. Kinetic diagram ma x L/ mg B P

10 Free Body Diagram and Kinetic Diagram The ladder AB lide down the wall a hown. The wall and floor are both rough. Draw the FBD and KD for the ladder.

11 Free Body Diagram and Kinetic Diagram 1. Iolate body. Axe 3. Applied force 4. Replace upport with force 5. Dimenion 6. Kinetic diagram q N B ma y F B I = ma x W y N A F A x

Sample Problem 16.1 At a forward peed of 30 ft/, the truck brake were applied, cauing the wheel to top rotating. It wa oberved that the truck to kidded to a top in 0 ft.

12 Sample Problem 16.1 At a forward peed of 30 ft/, the truck brake were applied, cauing the wheel to top rotating. It wa oberved that the truck to kidded to a top in 0 ft. Determine the magnitude of the normal reaction and the friction force at each wheel a the truck kidded to a top. SOLUTION: Calculate the acceleration during the kidding top by auming uniform acceleration. Draw the free-body-diagram equation expreing the equivalence of the external and effective force. Apply the three correponding calar equation to olve for the unknown normal wheel force at the front and rear and the coefficient of friction between the wheel and road urface.

Sample Problem 16.1 v ft 30 x 0 0 ft SOLUTION: Calculate the acceleration during the kidding top by auming uniform acceleration. v v 0 0 30 a ft x x a 0 0ft a ft.

13 Sample Problem 16.1 v ft 30 x 0 0 ft SOLUTION: Calculate the acceleration during the kidding top by auming uniform acceleration. v v a ft x x a 0 0ft a ft.5 Draw a free-body-diagram equation expreing the equivalence of the external and inertial term. Apply the correponding calar equation. F y F y eff N A N B W 0 F x F x eff k F N A A F N W k B B k ma W a g g a

14 Sample Problem 16.1 Apply the correponding calar equation. M A M A eff N N N A W N W N F N 5ftW 1 ftn 4ft B B 1 1 W 5W 4 g 0.650W B 1 N rear 1 A N 1 N front 1 V W a W W W rear k rear 175 B ma W N rear W F rear 0. 1W N front 0. 35W Ffront k N front 35 F front W a g

15 Sample Problem 16. The thin plate of ma 8 kg i held in place a hown. Neglecting the ma of the link, determine immediately after the wire ha been cut (a) the acceleration of the plate, and (b) the force in each link. SOLUTION: Note that after the wire i cut, all particle of the plate move along parallel circular path of radiu 150 mm. The plate i in curvilinear tranlation. Draw the free-body-diagram equation expreing the equivalence of the external and effective force. Reolve into calar component equation parallel and perpendicular to the path of the ma center. Solve the component equation and the moment equation for the unknown acceleration and link force.

16 Sample Problem 16. SOLUTION: Note that after the wire i cut, all particle of the plate move along parallel circular path of radiu 150 mm. The plate i in curvilinear tranlation. Draw the free-body-diagram equation expreing the equivalence of the external and effective force. Reolve the diagram equation into component parallel and perpendicular to the path of the ma center. F t F t eff W co30 ma mg co30 a 9.81m/ co 30 a 8.50 m 60 o

17 Sample Problem 16. a 8.50 m 60 o Solve the component equation and the moment equation for the unknown acceleration and link force. M G M G eff FAE in 3050 mm FAE co30100 mm F in 3050 mm F co30100 mm 0 F DF DF 38.4 F F F F AE AE AE AE 11.6 F F AE F n F n eff F DF F DF 0 W in 30 0 AE 8kg9.81m DF W in 30 0 F AE 47.9 N T FDF N F DF 8.70 N C

18 Sample Problem 16.3 SOLUTION: Determine the direction of rotation by evaluating the net moment on the pulley due to the two block. Relate the acceleration of the block to the angular acceleration of the pulley. A pulley weighing 1 lb and having a radiu of gyration of 8 in. i connected to two block a hown. Auming no axle friction, determine the angular acceleration of the pulley and the acceleration of each block. Draw the free-body-diagram equation expreing the equivalence of the external and effective force on the complete pulley plu block ytem. Solve the correponding moment equation for the pulley angular acceleration.

19 Sample Problem 16.3 SOLUTION: Determine the direction of rotation by evaluating the net moment on the pulley due to the two block. note: M I G mk 10lb6in 5lb10in 10in lb rotation i counterclockwie. 1 lb 3. ft W g k lb ft ft Relate the acceleration of the block to the angular acceleration of the pulley. a A r A 10 1 ft a B r B 6 1 ft

Sample Problem 16.3 Draw the free-body-diagram equation expreing the equivalence of the external and effective force on the complete pulley and block ytem.

20 Sample Problem 16.3 Draw the free-body-diagram equation expreing the equivalence of the external and effective force on the complete pulley and block ytem. Solve the correponding moment equation for the pulley angular acceleration. M G M G eff lb ft 5lb ft I m 1 1 BaB ft m 1 AaA ft I a a lb ft A B 10 ft 1 6 ft 1 Then, a A r A a B r B 10 ft.374 rad 1 6 ft.374 rad rad a A a B 1.978ft ft

21 Sample Problem 16.4 A cord i wrapped around a homogeneou dik of ma 15 kg. The cord i pulled upward with a force T = 180 N. Determine: (a) the acceleration of the center of the dik, (b) the angular acceleration of the dik, and (c) the acceleration of the cord. SOLUTION: Draw the free-body-diagram equation expreing the equivalence of the external and effective force on the dik. Solve the three correponding calar equilibrium equation for the horizontal, vertical, and angular acceleration of the dik. Determine the acceleration of the cord by evaluating the tangential acceleration of the point A on the dik.

22 Sample Problem 16.4 SOLUTION: Draw the free-body-diagram equation expreing the equivalence of the external and effective force on the dik. Solve the three calar equilibrium equation. F x F x eff 0 ma x 0 F y F y eff T W a y T ma W m y 180 N - M G M G eff Tr I T mr 1 mr 15kg9.81m 15kg 180 N 15kg0.5m a y.19 m a x 48.0rad

23 Sample Problem 16.4 Determine the acceleration of the cord by evaluating the tangential acceleration of the point A on the dik. a cord a a a A t.19 m A G t 0.5m48 rad a cord 6.m a x 0 a y.19 m 48.0rad

24 Prob # 16.5 A uniform rod BC of ma 4 kg i connected to a collar A by a 50-mm cord AB. Neglecting the ma of the collar and cord, determine (a) the mallet contant acceleration aafor which the cord and the rod will lie in a traight line, (b) the correponding tenion in the cord.

Prob# 16.14 A uniform rectangular plate ha a ma of 5 kg and i held in poition by three rope a hown.

25 Prob# A uniform rectangular plate ha a ma of 5 kg and i held in poition by three rope a hown. Knowing that θ= 30, determine, immediately after rope CF ha been cut, (a) the acceleration of the plate, (b) the tenion in rope AD and BE.

Prob # 16.18 The 15-lb rod BC connect a dik centered at A to crank CD.

26 Prob # The 15-lb rod BC connect a dik centered at A to crank CD. Knowing that the dik i made to rotate at the contant peed of 180 rpm, determine for the poition hown the vertical component of the force exerted on rod BC by pin at B and C.

Prob # 16.38 Dik A and B are bolted together, and cylinder D and E are attached to eparate cord wrapped on the dik. A ingle cord pae over dik B and C.

27 Prob # Dik A and B are bolted together, and cylinder D and E are attached to eparate cord wrapped on the dik. A ingle cord pae over dik B and C. Dik A weigh 0 lb and dik B and C each weigh 1 lb. Knowing that the ytem i releaed from ret and that no lipping occur between the cord and the dik, determine the acceleration (a) of cylinder D, (b) of cylinder E

ME 141. Lecture 7: Friction

ME 141. Lecture 7: Friction ME 141 Engineering Mechanic Lecture 7: riction Ahmad Shahedi Shail Lecturer, Dept. of Mechanical Engg, BUET E-mail: hail@me.buet.ac.bd, hail6791@gmail.com Webite: teacher.buet.ac.bd/hail Courtey: Vector