b mech engg 6-dynamics

8/18/2019 B Mech Engg 6-Dynamics

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Chapter 18- Planar Kinetics of aRigid Body: Work and Energy

STATICS and DYNAMICS- 11th Ed.,

R. C. Hibbeler and A. GuptaCoure Intru!tor" Mi Sa#an Shahid


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Objective:

We will apply work and energy methods toproblems involving force, velocity, and

displacement related to the planar motion of a

rigid body.


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KineticEnergyCalclating rigid body!s KE when it is

sbjected to

"i# translation,

"ii# rotation abot a $%ed a%is, or

"iii# general plane motion.

Consider the rigid body shown in inertial%&y plane. 'n arbitrary ith particle ofthe body, having mass dm, is locatedat r from the arbitrary point (.

)ere *"+# is the moment of inertia for thebody abot an a%is which isperpendiclar to the plane of motion

and passes throgh the mass center.


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&-ranslationWhen a rigid body of mass m

is sbjected to either

rectilinear or translation, theKE de to rotation is ero,since / 01

where v"+# is the magnitdeof the translational velocity vat the instant considered.


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'%is

When a rigid body is rotating abota $%ed a%is passing throgh pointO, the body has both translationaland rotational KE.

-he body!s KE may also beformlated by noting thatv(G)=r(G)ω

by the (arallel&'%is theorem, the

terms inside the parenthesesrepresent the moment of inertia*"O# of the body abot an a%isperpendiclar to the plane ofmotion and passing throgh pointO.


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5&+eneral (lane 6otionWhen a rigid body is sbjected to

general plane motion, it has an anglarvelocity / and its mass center has avelocity v"+#.

-otal KE of the body consists of thescalar sm of the body!s translationalKE and rotational KE abot its masscenter.

-he e7ation can also be e%pressed interms of body!s motion abot itsinstantaneos center of ero velocity,

Where *"*C# is the moment of inertial ofthe body abot its instantaneos

center.


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Example


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-he Work of a 4orce8y variable force

8y constant force

8y weight8y a spring force


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# Work of a 9ariable

4orce*f an e%ternal force 4

acts on a rigid body,the work done by the

force when it movesalong the path s.

)ere is the anglebetween the tails of theforce vector and thedi;erential

displacement.


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2# Work of a Constant

4orce*f an e%ternal force 4"c#

acts on a rigid body,and maintains a

constant magnitdeand constant direction,while the bodyndergoes a translation

s.


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5# Work of a Weight -he weight of a body does

work only when the body!scenter of mass + ndergoes

a vertical displacement


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=# Work of a

>pring 4orce*f linear elastic spring is attached to a body,

the spring force F(s)=ks, acting on the bodydoes work when the spring either stretches orcompresses from s to a frther position s2.

*n both cases, the work will be negative sincethe displacement of the body is in theopposite direction to the force.

Where ?s2?@?s?


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4orce -hat Ao Bot Work -here are some e%ternal forces that do not work when

the body is displaced.

-here forces can act either at $%ed points on the body, orthey can have a direction perpendiclar to theirdisplacement.

E%amples:

1- Reactions at a Pin support about which a body

rotates2- Normal reaction acting on a body that moves along

a fxed surace.

3- Weight o a body when the center o gravity o thebody moves in a horizontal plane.


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xplanation!' frictional force 4"f# acting on a rond

body as it rolls without slipping over arough surace also does no work.

-his is becase, dring any instant oftime dt, 4"f# acts at a point on the bodywhich has ero velocity "instantaneoscenter *C# and so the work done by theforce on the point is ero.

*n other words, the point is notdisplaced in the direction of the forcedring this instant. >ince 4"f# contactssccessive points for only an instant,the work of 4"f# will be ero.


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-he Work of a CopleWhen a body sbjected to a cople

ndergoes general plane motion,the two cople forces do work onlywhen the body ndergoes arotation.

Consider the body which is

sbjected to a cople momentM=Fr

When the body translates, schthat the component of

displacement along the line ofaction of the forces is ds"t#.

-he positive work of one forcecancels the negative work of theother.


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Contd.*f the body ndergoes a di;erential rotation dθ

about an axis which is perpendicular to the plane of the

couple and intersects the plane at point O,

Then, each force undergoes a displacement ds(θ)=(r/2)

dθ in the direction of the force.

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