Usaha Dan Energi(KUL4)

7/24/2019 Usaha Dan Energi(KUL4)

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Work and Energy


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Work


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Work

The work done by a constantforce on an object that is undergoing astraight linedisplacement is given by

Definition of work is based on observations. You do work by exerting the

forceon an object while that object movesfrom one place to another(undergoes displacement)

You do more work if the force is greater

You do more work if displacement is greater

SFW =


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Work

SIunit of work isJoule (J) !Jewel"

#Joule$ (# Newton) (# meter)% #J$ # Nm

Britishunit of work is foot-pound(ft&lb)

'nit of force is pound unit of distance is foot

onversion #J$ *.+,+- ftlb # ftlb$ #.,-J

SFW =

James Joule

#/#/ 0 #//1


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Work

You push a stalled car through a displacement Swith a constant force Finthe direction of motion

You push a stalled car through a displacement Swith a constant force Fat

angle to the direction of motion

2nly component of force in direction of car3s displacement is important

SFW =

cosSFW =


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Work and Kinetic Energy


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4rom the definition of workwe know that the total work done on anobject is related to its displacement(changes in position).

Workis also related to changesin thespeedof the object.

SFW =


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Example 5lock sliding on a frictionlesstable

4orces acting on a block its weight normalforce and the force Fexerted by the hand.


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6. The netforce on a block isin the direction of its

motion. 4rom N! this

means that the block

speeds up. W$"&Salso

tells us that the total workwill be positive.

5. 7ere only the component"&cos contributes to Wtotal.

The block speeds up as well.


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. The netforce here opposesthe displacement. 4rom

N! this means that theblockslows down. W$"&Salso tells us that the totalwork will be negative.

D. 7ere the netforce is#eroso the speed of the blockstays the same and Wtotalis

8ero.


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9hen an object undergoes a displacement

object will :speed up: if Wtotal>*

object will :slow down: if Wtotal


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Savv x+= 22

1

2

2

onsider a particle with mass mmoving along thex0axis under the action ofa constant netforce with magnitude "directed along the positivex0axis.

;article acceleration is constant and by N! "$m&ax.

uation

S

vvax

2

2

1

2

2 =

S

vv

mmaF x 2

2

1

2

2

==

2

1

2

2 2

1

2

1

mvmvSF =

12 xxS =


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The product "&Sis the work done by the netforce. Thus it is e>ual to the

total work Wtotaldone by all the forces acting on a particle.

Definition of &inetic Energ'

?ike work kinetic energy of a particle is ascalar>uantity it depends on

particle3s mass and speed notits directionof motion. ar has the same kinetic energy when going north at #*m/sas

when going east at #*m/s.

@inetic energy can neverbe negative% it3s#erowhen particle is at rest.

2

2

1mvK=

2

1

2

22

1

2

1mvmvSF =


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Work - Energy Theorem

Work done b' the net force on a particle e(uals the change in theparticle)s kinetic energ'*

Work - Energ' +heorem

2

21

ii mvK =21

22

21

21 mvmvSF =

KKKWtot == 12

12 KKSF =

9hen an object moves

object will :speed up: if Wtotal>* &=A

object will :slow down: if Wtotal


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Kinetic Energy

The example with the hammerhead gives insight into the physicalmeaning of kinetic energy.

The hammerhead was dropped from rest and its kinetic energy when ithits the 0beam e>uals the total work done on it up to that point by thenet force.

To accelerate a particle with mass mfrom rest (8ero kinetic energy) upto a speed v the total work done on it must e>ual the change in kineticenergy from 8ero to &$*.&m&v=

&inetic energ' of a particle is e(ual to the total work that was

done to accelerate it from rest to its present speed. 2r from its present speed to restC

atch the ball right pull your hand back increasing distance to stopthe ball ball does the work on your hand e>ual to the ball3s initial kineticenergy Wtot$"


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Composite Systems

Ean standing on frictionless roller skates on a levelsurface pushes against the rigid wall setting himself inmotion to the right.

4orces acting on him his weight W upward normalforces n$and nexerted by the ground on his skates

and the hori8ontal force "exerted on him by the wall.

No verticaldisplacement so forces W n$and ndoN,work. 4orce "exerted on him by the wall ishori8ontal force that accelerates him to the right but hishands don3t move.


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Work andEnergy withVarying

Forces


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Work andEnergy withVarying Forces

9e know that work done by a constantforceon an object that isundergoing astraight linedisplacement is given by

9hat happens when force exerted on an object is N,+ constantand the object moves in path which is N,+ straightG

Hxample spring stretched

Eore you stretch it the harder you have to pull thus theforce is non-constant

?et3s considerstraight0line motion with non-constantforce

2ne complication at a timeC

"xchange along thex0axis (force depends on position)

SFW =


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;article moves fromx#tox=% "xdepends on coordinatex. ?et3s divide the total displacement by small segments xa xb xcI Total work done during segment xa J by the average force "ain this

segment multiplied by the displacement xa. 6ll segments

=+++=

mmffbbaa xFxFxFxFW ...


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f number of segments is very large segmentKs width xis very small n the limit thesumis integralof "xfrom x#to x=.

( ) == 2

1

0lim

x

x

xmmx

dxFxFW

2n a graph of force as function of

position the total workdone by this

force is represented by the area

under the curvebetween the initial

and final positions.

Farying x0component of forcestraight0line displacement


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?et3s check it if "xis constantfrom x#to x=.

SFxxFdxFdxFW x

x

x

x

x

x

x ==== )( 122

1

2

1


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Hookes Law

To keep idealspringstretched by amountxbeyond its initial length weneed to apply on the end the forcewhich isproportionaltox

xkFx = 4orce re>uired to stretch a spring

KL spring (force) constant MNOmP

Hxample 4orce constant k

4loppy toy spring k$# N/m ar suspension spring k$#*N/m

Robert Hooke

#-, L #+*,

... lean, bent and ugly man ...


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Hookes Law Tostretcha spring you must do work


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Hookes Law

ompression

4orce "xand displacementxare both negative

4orce is in thesame directionas displacement work ispositive

Example is following/

What happens if 'ou compress the spring

xF

x x


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Varying Forces: Work - EnergyTheorem

=== 2

1

2

1

2

1

x

x

xx

x

x x

x

x xtot dx

dx

dvmvdxmadxFW

2ne can use the same approach divide total displacement into segments 6pply 9ork0Hnergy Theorem for each segment Wa$@a$"axa


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Cr!ed "ath: Work - Energy Theorem

"orcethat variesin direction and magnitude

%isplacementlies along a curved path particle moves from ;#to ;=

=== 2

1

2

1

2

1

cosP

P

P

P

P

PdFdFdFW

Divide curve between ;#and ;=into small vector displacements d.

Hach d is a tangent to the path at its position.

"is the force at a point along the path is the angle between "and d.


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"ower

!;ower man"

Er. 2lympia

9att3s

Hngine


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"ower

You lift weight #**Nvertically at a distance #m at constant velocity You do (#**N)(#m)$#**Jof work whether it takes # sec # hour # yearI

You want to know how (uickl'the workis done

0oweris the time rateat which work is done. ;ower is a scalar.

6verage power

nstantaneous power

%efinition of work makes no reference to the passage of time

t

WPav

=

dt

dW

t

WP

t=

=

0lim


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"ower

The Sunit of power is !att(9) # 9 $ # Roule per # second.

n the 5ritish system of units power is in ft&lbOsec or in a larger unit called horsepower(hp). # hp $ * ft&lbOsec $ ,,*** ft&lbOmin $ +S- 9 $ of kilowatt (k9)

The kilowatt0hour (k9h) is a unit of electrical energy notpower

energy $ power & time

avav vFtSF

t

SF

tWP ====

Fvdt

dW

t

WP

t==

=

0lim vFP =

n terms of scalar product

n mechanicspoweris expressed

in terms of forceand velocit'

#ames Watt

$%&' ( $)$*

Watt's steam engines

Started with nothing,

died as a very wealthyman


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"otentia+ Energy and Energy

Conser!ation


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Warm-,p: "ower

"o!ercl#mb Uunner with mass mruns up the stairs to the top of SS,0m0tall


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ra!itationa+ "otentia+

Energy


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ra!itationa+"otentia+ Energy


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ra!itationa+"otentia+ Energy

Hnergy associated with position is calledpotential energ'

f elevation for which the gravitational potential energy is chosen to be8ero has been selected then the expression for the gravitationalpotential energy as a function of position 'is given by

1ravitational potential energ'2$ravis associated with the work done

by the gravitational force according to

mgyUgrav=

UUUUUWgrav === )( 1221


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Conser!ati!ewithNon-Conser!ati!e

Forces


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Conser!ati!e and.on-Conser!ati!e

Forces

9ork done by the conservativeforce only depends on the initialand finalpositions and doesn3t depend on the path

Uunner gravitational force is conservative

4rom point # to point = same work

The work done by a conservativeforcehas these properties

t can always be expressed as the differencebetween the initial and final values of apotential energ'function 2$ 0W.

t is reversible.

t is independentof thepathof the body anddepends only on the starting and ending points.

9hen the initial and final points are the same(closed loop the total work is#ero.

!ll fo"ces which do not satisfy these p"ope"ties a"enon-conservative fo"ces.


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E+astic "otentia+ Energy


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E+astic"otentia+ Energy

9hen you compress a spring

f there is no friction spring moves back

@inetic energy has been !stored" in the

elasticdeformation of the spring

Uubber0band slingshot the same principle

9ork is done on the rubber band by the

force that stretches it

That work isstoredin the rubber band

until you let it go

You let it go the rubber gives kinetic

energy to the projectile

Elasticbody if it returns to its original

shape and si8e afterbeing deformed


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EquilibriumSpring is stretched/t does negati!e work on 0+ock

Spring relaxes/t does positi!e work on 0+ock

Spring is compressed

"ositi!e work on 0+ock

1+ock mo!es 2rom one position 3$to another position 34: how

much workdoes the elastic5spring6 2orce do on the 0+ock7


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Work done ,N a springto move one endfrom elongationx#to a different elongation

x=

9hen we stretch the spring we dopositive workon the spring

9hen we relax the spring work done on

the springis negative

9ork done B3the spring

4rom N4! >uantities of work arenegatives of each other

Thus work Weldone by the spring

9e can express the work done B3thespring in terms of a given >uantity at thebeginning and end of the displacement [ ]JkxU 2

2

1=

2

1

2

22

1

2

1kxkxW =

2

2

2

12

1

2

1kxkxWel =

Elastic potential energy


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2

2

1 kxU= The graph of elastic potential energy

for idealspring is a parabola

4or extensionof springx56

4orcompressionx76

Hlastic potential energy 2is NE8E9negativeC

n terms of the change of potentialenergy

2

2

2

1

21

2

1

2

1kxkx

UUUWel

====


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9hen a stretched spring is stretched greater Welis negative and2increases greater amount of elastic potential energy is storedin the spring

9hen a stretched spring relaxesxdecreases Welis positive and

2decreases spring loses its elastic potential energy

Eore spring compressed ,9stretched greater its elasticpotential energy

2

2

2

1212

1

2

1kxkxUUUWel ===


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E+astic"otentia+ Energy:Work - EnergyTheorem

12 UUWW eltot == 9ork L Hnergy Theorem Wtot$&=0 no matter

what kind of forces are acting on the body. Thus

22111221 UKUKKKUUWtot +=+== /2 on+y e+astic 2orcedoes work

22

22

21

21

21

21

21

21 kxmvkxmv +=+

Total mechanical energy E(the sum of elastic potential energyand kinetic energy) is conserved

deal spring is frictionlessand massless

f spring has a mass it also has kinetic energy

Your car has a mass of #.= ton or more


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E+asticForce8 other 2orces7

12 KKWWW othereltot =+=

f forces other than elastic force also dowork on the body the total work is

2211 UKWUK other +=++ e+astic 2orce 8 other 2orces

2

2

2

2

2

1

2

1 2

1

2

1

2

1

2

1

kxmvWkxmv other +=++ The work done by all forces other than the elastic force

e>uals the change in the total mechanical energy E ofthe system where ' is the elastic potential energy

!


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Force and"otentia+ Energy


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Force and "otentia+ Energy

9e have studied in detail two specific conservativeforces gravitational force and elastic force.

9e have seen there is a definite relationship between aconservative force and the corresponding potential

energy function. The force on a mass in a uniform gravitational field is

"'$ 0 mg. The corresponding potential energy function

is 2(y) $ mgy.

The force exerted on a body by a spring of forceconstant kis "x$ 0 kx. The corresponding potential

energy function is 2s(x) $ (#O=)kx=

. n some situations you are given an expression for

potential energy as a function of positionand have tofind corresponding force.


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onsider motionalong astraight line with coordinatex

"x(x) is thex0component of force as function ofx

2(x) is the potential energy as function ofx

9ork done by conservative force e>uals the negative of the change

2in potential energy UW =

4or infinitesimal displacement x the work done by force "x(x) duringthis displacement is J "x(x)x(suppose that this interval is so small thatthe force will vary just a little)

n the limit x*

UxxFx =)( x

U

xFx

=)(

dx

xdUxFx

)()( =

4orce from potentialenergy one dimension


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n regions where2(x) changes most rapidly with x (i.e. where d2(x)Odxislarge) the greatest amount of work is done during the displacement andit corresponds to a large force magnitude

9hen "x(x) is inpositivex0direction 2(x) decreaseswith increasingx

Thus "x(x) and 2(x) have opposite sign

Thus the forceis proportional to the negative slope of thepotentialenerg'function

The physical meaning conservative force alwa's acts to push thes'stem toward lower potential energ'

dx

xdUxFx

)()( =

4orce from potentialenergy one dimension


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?ets verify if this expression correctly gives the gravitational force and the elastic forcewhen using the gravitational potential energy and the elastic potential energy

2

2

1)( kxxU = kxkx

dx

d

dx

xdUxFx =

== 22

1)()(

mgyyU =)( ( ) mgmgydyd

dy

ydU

xFy ===

)(

)(

The gravitational potential energy is linearly related to theelevation (i.e. constant slope) and the force is constant.

The elastic potential energy varies >uadratically with position.The force varies in a linearly.


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Energy 9iagrams


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Energy 9iagrams

n situations where a particle moves inone0dimension only under influence of asingle conservative force it is very usefulto study the graph of the potentialenergy as a functionof position 2(x)

6t any point on a graph of 2(x) theforcecan be calculated as the negative

of theslope of the potential energ'function

"x$ 0 d2Odx

Example Qlider on an air track

Usaha Dan Energi(KUL4)

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