Plane Motion Of Rigid Bodies: Forces And Accelerations

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Plane Motion of RigidBodies:Forces andAccelerationsReference:Beer, Ferdinand P. et al, Vector Mechanics for Engineers : Dynamics, 8th Edition, Mc GrawHillHibbeler R.C., Engineering Mechanics: Dynamics, 11th Edition, Prentice Hall (Chapter 17)1

Introduction In this chapter and in Chapters 17 and 18, we will beconcerned with the kinetics of rigid bodies, i.e., relationsbetween the forces acting on a rigid body, the shape and massof the body, and the motion produced. Results of this chapter will be restricted to:- plane motion of rigid bodies, and- rigid bodies consisting of plane slabs or bodies whichare symmetrical with respect to the reference plane. Our approach will be to consider rigid bodies as made oflarge numbers of particles and to use the results of Chapter14 for the motion of systems of particles. Specifically,rrr&rF maandM H GG D’Alembert’s principle is applied to prove that the externalrforces acting on a rigid body are equivalent a vector maattached to the mass center and a couple of moment I α .2

Equations of Motion for a Rigid Body3

Angular Momentum of a Rigid Body in Plane Motion4

MOMENT OF INERTIAThe mass moment of inertia is a measure of anobject’s resistance to rotation.I r2 dm r2ρ dV5

PARALLEL-AXIS THEOREM6

Plane Motion of a Rigid Body7

EQUATIONS OF TRANSLATIONAL MOTION 8

EQUATIONS OF MOTION: TRANSLATION ONLY9

EQUATIONS OF MOTION: TRANSLATION ONLY10

PROCEDURE FOR ANALYSISProblems involving kinetics of a rigid body in only translationshould be solved using the following procedure.1. Establish an (x-y) or (n-t) inertial coordinate system and specifythe sense and direction of acceleration of the mass center, aG.2. Draw a FBD and kinetic diagram showing all external forces,couples and the inertia forces and couples.3. Identify the unknowns.4. Apply the three equations of motion:Σ Fx m(aG)xΣ Fy m(aG)yΣ Fn m(aG)n Σ Ft m(aG)tΣ MG 0 orΣ MP Σ (Mk)P Σ MG 0 or Σ MP Σ (Mk)P5. Remember, friction forces always act on the body opposing themotion of the body.11

Sample Problem 16.1At a forward speed of 30 m/s, the truck brakes wereapplied, causing the wheels to stop rotating. It wasobserved that the truck to skidded to a stop in 200 m.Determine the magnitude of the normal reaction and thefriction force at each wheel as the truck skidded to a stop.12

Sample Problem 16.113

Sample Problem 16.2The thin plate of mass 8 kg is held in place as shown.Neglecting the mass of the links, determine immediatelyafter the wire has been cut (a) the acceleration of theplate, and (b) the force in each link.14

15

EXAMPLEGiven:A 50 kg crate rests on a horizontalsurface for which the kinetic frictioncoefficient µk 0.2.Find:The acceleration of the crate if P 600 N.16

EXAMPLE (continued)17

GROUP PROBLEM SOLVINGGiven: A uniform connecting rod BC has amass of 3 kg. The crank is rotatingat a constant angular velocity of ωAB 5 rad/s.Find:The vertical forces on rod BC atpoints B and C when θ 0 and 90degrees.18

EQUATIONS OF MOTION FOR PURE ROTATION19

EXAMPLEGiven:Find:A rod with mass of 20 kg is rotating at 5rad/s at the instant shown. A momentof 60 N·m is applied to the rod.The angular acceleration α and thereaction at pin O when the rod is in thehorizontal position.20

EXAMPLE21

Sample Problem 16.3A pulley weighing 12 N and having a radius of gyration of 8cm is connected to two blocks as shown.Assuming no axle friction, determine the angular accelerationof the pulley and the acceleration of each block.22

23

EQUATIONS OF MOTION: GENERAL PLANE MOTIONWhen a rigid body is subjected to externalforces and couple-moments, it canundergo both translational motion as wellas rotational motion. This combination iscalled general plane motion.Using an x-y inertial coordinatesystem, the equations of motions aboutthe center of mass, G, may be writtenas Fx m (aG)xP Fy m (aG)y MG I G α24

Sample Problem 16.4A cord is wrapped around a homogeneous disk ofmass 15 kg. The cord is pulled upwards with a force T 180 N.Determine: (a) the acceleration of the center of thedisk, (b) the angular acceleration of the disk, and (c)the acceleration of the cord.25

Sample Problem 16.426

Constrained Plane Motion: Rolling Motion For a balanced disk constrained toroll without sliding,x rθ a r α Rolling, no sliding:F µs Na rαRolling, sliding impending:F µs Na rαRotating and sliding:F µk Na, rα independent For the geometric center of anunbalanced disk,a O rαThe acceleration of the mass center,rrraG aO aG Orrr aO aG O aG O()t ()n27

Sample Problem 16.6mE 4 kgk E 85 mmmOB 3 kgThe portion AOB of the mechanism is actuatedby gear D and at the instant shown has aclockwise angular velocity of 8 rad/s and acounterclockwise angular acceleration of 40rad/s2.Determine: a) tangential force exerted by gear D,and b) components of the reaction at shaft O.28

Sample Problem 16.6mE 4 kgk E 85 mmα 40 rad s 2ω 8 rad/smOB 3 kg29

Sample Problem16.8SOLUTION: Draw the free-body-equation for thesphere, expressing the equivalence ofthe external and effective forces. With the linear and angular accelerationsrelated, solve the three scalar equationsderived from the free-body-equation forthe angular acceleration and the normalA sphere of weight W is released withand tangential reactions at C.no initial velocity and rolls without Calculate the friction coefficient requiredslipping on the incline.for the indicated tangential reaction at C.Determine: a) the minimum value Calculate the velocity after 10 m ofof the coefficient of friction, b) theuniformly accelerated motion.velocity of G after the sphere has Assuming no friction, calculate the linearrolled 10 m and c) the velocity ofacceleration down the incline and theG if the sphere were to move 10 mcorresponding velocity after 10 m. 30down a frictionless incline.

Sample Problem 16.9A cord is wrapped around theinner hub of a wheel and pulledhorizontally with a force of 200 N.The wheel has a mass of 50 kgand a radius of gyration of 70 mm.Knowing µs 0.20 and µk 0.15,determine the acceleration of G andthe angular acceleration of the wheel.31

Sample Problem 16.932

Sample Problem 16.933

Sample Problem 16.10The extremities of a 4-m rodweighing 50 N can move freely andwith no friction along two straighttracks. The rod is released with novelocity from the position shown.Determine: a) the angularacceleration of the rod, and b) thereactions at A and B.34

Sample Problem 16.1035

Sample Problem 16.1036

- plane motion of rigid bodies, and - rigid bodies consisting of plane slabs or bodies which are symmetrical with respect to the reference plane. D’Alembert’s principle is applied to prove that the external forces acting on a rigid body are equivalent a vector attached to the mass center and a couple of moment ma r Iα.

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