4 Work and Energy

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AS-evel !aths"!echanics #for Edexcel

M2.4 Work and

Energy


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$onte

nts


%nerg&

%nerg&

'or(

$onservation of energ&

'or(-energ& and dissipative forces

Power

%)amination-st&le *uestions


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!echanical energ&

Energy is measured in+oulesor (ilo+oules, reresented

!y " or k".

!echanical energ&is energy t#at a arti$%e #as due to

eit#er its motion or its osition.

Energy e&ists in many forms. '#ese in$%ude e%e$tri$a%

energy, $#emi$a% energy, #eat energy and %ig#t energy.

Me$#ani$a% energy is t#e energy t#at wi%% !e

$onsidered in M2. (t $omes in two forms ) (inetic

and potentialenergy.

%nerg&is a measure of a arti$%es $aa$ity to do work.


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inetic and potential energ&

inetic %nerg&

*ineti$ energy is t#e energy a arti$%e ossesses due to itsmotion.

*ineti$ Energy is defined as

w#ere mis mass in kg and vis seed in ms+1.

ravitational Potential %nerg&

ra-itationa% otentia% Energy is t#e energy a arti$%e

ossesses due to its position. (t is stored energy.

/#ange in ra-itationa% otentia% Energy is defined as

w#ere mis mass in kg,gis a$$e%eration due to gra-ity and his

t#e -erti$a% distan$e tra-e%%ed in metres.

mv2

mgh,


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inetic energ&

ind t#e kineti$ energy of t#e fo%%owing

a 5 arti$%e of mass 0. kg tra-e%%ing at ms+1

! 5 $ar of mass 700 kg tra-e%%ing at 1 ms+1

$ 5 !u%%et of mass 20 g tra-e%%ing at 00 ms+1

a *.E., in 8ou%es, 9 : m : -2

9 : 0. : 2

9 /

! *.E., in 8ou%es, 9 : m : -2

9 : 700 : 1

2

9 0 #10

$ *.E., in 8ou%es, 9 : m : -2

9 : 0.02 : 002

9 #100


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ind t#e gra-itationa% otentia% energy of t#e fo%%owing, stating

w#et#er it is a gain or %oss in gra-itationa% otentia% energy

ravitational potential energ&

a ..E., in 8ou%es, 9 m:g: h

9 2 : 7. :

9 1/.2 '#is is a gain in ..E.

! ..E., in 8ou%es, 9 m:g: h

9 00 : 7. : 0

9 #31 000 '#is is a %oss of ..E.

$ ..E., in 8ou%es, 9 m:g: h

9 0 : 7. : ;

9 12 200 '#is is a gain in ..E.

a 5 arti$%e of mass 2 kg raised a -erti$a% distan$e of m

! 5 %ift of mass 00 kg des$ending 0 m -erti$a%%y

$ 5 man of mass 0 kg $%im!ing a -erti$a% distan$e of ; m.


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%nerg& *uestion

4uestion ind t#e gain in kineti$ energy of a $ar of mass

00 kg as it a$$e%erates from 10 ms+1

to 2 ms+1

.

(nitia% *.E. 9 : m: v2

9 : 00 : 102

9 40 000

ina% *.E. 9 : m: v2

9 : 00 : 22

9 20 000

'#erefore t#e gain in kineti$ energy of t#e $ar is #0 0005.


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%nerg& *uestion #

4uestion # 5 $#i%d of mass 30 kg s%ides down a s%ide of

%engt# 4 m in$%ined at an ang%e of 30< to t#e #ori=onta%. Bymode%%ing t#e $#i%d as a arti$%e and t#e s%ide as a smoot#

in$%ined %ane, find t#e %oss of gra-itationa% otentia% energy

of t#e $#i%d.

30

..E. 9 m:g: h

9 30 : 7. : 4sin30

@ire$tion of motion

2 kg


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'or( *uestion 6

4uestion 6 5 stone of mass 0 g fa%%s from t#e to of

a $%iff of #eig#t ; m. ind t#e work done !y gra-ity ast#e stone fa%%s.

0.0g >

Work done 9 for$e : distan$e

9 ?0.0 : 7. : ;

9 36.;

'#erefore t#e work done !y gra-ity is 6/.71 5.



'or( *uestion 3

4uestion 3 5 arti$%e of mass kg is u%%ed m u a

roug# %ane in$%ined at an ang%e of 60< to t#e #ori=onta%.'#e $oeffi$ient of fri$tion !etween t#e %ane and t#e

arti$%e is 0.3.

60

a ind t#e work done against fri$tion.

! ind t#e work done against gra-ity.



'or( *uestion 3

a 'o find t#e work done against t#e fri$tiona% for$e it is first

ne$essary to find t#e fri$tiona% for$e.Aeso%-ing erendi$u%ar to t#e %ane

R9 g$os60< 9 2.g

F9R

9 0.3 : 2.g9 ;.3

'#e fri$tiona% for$e is ;.3 >.

'#e distan$e tra-e%%ed in t#e dire$tion oosite to t#e

fri$tiona% for$e is m.

Work done against fri$tion 9 ;.3 : 9 12.2 5.



'or( *uestion 3

! 'o find t#e work done against gra-ity it is ne$essary to

$a%$u%ate t#e -erti$a% distan$e tra-e%%ed.

Work done against gra-ity 9 g: sin60ewtonFs e$ond Law.

5 arti$%e of mass mkg is mo-ed a distan$e of

dm #ori=onta%%y !y a $onstant #ori=onta% for$eF>.

Let t#e initia% seed !e ums+1

and t#e fina% seed !evms+1.

Work @one 9 /#ange in *ineti$ Energy



Remember:

s= distance

u= initial velocity

v= final velocity

'or(-energ& principle

By >ewtonFs e$ond Law

F9 maa9FGm

'#is gi-es Fd9 mv2) mu2

in$e t#e for$e is assumed $onstant, t#e a$$e%eration

$an a%so !e assumed $onstant.Hsing v29 u2D 2as

v29 u2D 2 :FGm : d

2Fd

mv2

= u2

D

mv29 mu2DFd



'or(-energ& *uestion

4uestion 5 arti$%e of mass 3 kg is u%%ed 6 m !y a

#ori=onta% for$e of 10 > a$ross a smoot# #ori=onta% surfa$e.(f t#e arti$%e started from rest, find its seed w#en it #as

tra-e%%ed 6 m.

10 >3 kg

@ire$tion of motion

Work done 9Fd9 10 : 6 9 60

Fd9 mv2) mu2

Work done 9 mv2) mu29 60

: 3 : v2) 0 9 60

1.v29 60

v9 6.32 ?3 s.f.

'#erefore t#e seed of t#e arti$%e w#en it #as

tra-e%%ed 6 m is /.6# ms-.


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'or(-energ& *uestion #

4uestion # ind t#e #ori=onta% for$e t#at $auses a arti$%e

of mass 2. kg to in$rease seed from 2 ms+1

to ms+1

o-era distan$e of 10 m on a smoot# #ori=onta% surfa$e.

Work done 9 $#ange in kineti$ energy

(nitia% *.E. 9 : 2. : 229

ina% *.E. 9 : 2. : 29 31.2

/#ange in kineti$ energy 9 26.2

Work done 9Fd9 10F9 26.2F9 2.62

'#erefore t#e reIuired for$e is #./#1 N.

' ( d di i ti f



'or( and dissipative forces

Work done against a dissipative forceis eIua% to t#e %oss of

t#e tota% me$#ani$a% energy of t#e system.

5 dissipative forceis a for$e w#i$# $auses a arti$%e to

%ose energy.

ri$tion and air resistan$e are e&am%es of dissiati-e for$es.

'#is means t#at t#e rin$i%e of $onser-ation no %onger

a%ies, as t#e tota% me$#ani$a% energy of t#e arti$%e is

not $onstant.

' ( d di i ti f



'or( and dissipative forces

%)ample 5 arti$%e of mass 3 kg is mo-ing down a roug#

s%oe in$%ined at an ang%e of 30< to t#e #ori=onta%. '#earti$%e asses a ointAat 3 ms+1and a ointB2 m down

t#e s%oe at 4 ms+1. /a%$u%ate t#e work done against fri$tion.

ain in *.E. 9 : 3 : 42) : 3 : 32

9 10. "

Loss in ..E. 9 3 :g: 1 9 27.4 "

sin30< 9 h 2

Loss of energy 9 27.4 ) 10. 9 1.7

Work done against fri$tion 9 2.8 5

h2 m

30, find t#e ower rodu$ed !y t#e $y$%ist.

in$e t#e seed is $onstant t#ere is no a$$e%eration.

'#erefore, dri-ing for$e 9 resistan$e for$e.

@ri-ing for$e 9 20 >

Ke%o$ity 9 ; ms+1

ower 9 20 : ; 9 30 '

%)amination st&le *uestions


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$ontents



%nerg&

'or(

$onservation of energ&

'or(-energ& and dissipative forces

Power


%)am *uestion


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Boardworks Ltd 20063; of 46

%)am *uestion

%)amination-st&le *uestion 5 $ar of mass 1000 kg

mo-es a%ong a straig#t, #ori=onta% road. '#e $ar engine isworking at a $onstant rate of 4 kW and t#e tota% resistan$e

to t#e motion of t#e $ar is 00 >.

a ind t#e a$$e%eration of t#e $ar w#en its seed is 1 ms +1.

'#e $ar $omes to a #i%% w#i$# is in$%ined at an ang%e of to t#e #ori=onta%, w#ere sin9 0.1. '#e resistan$e to t#e

motion of t#e $ar is un$#anged.

! ind t#e ma&imum seed of t#e $ar u t#e #i%% w#en it

is working at a rate of 4 kW.

%)am *uestion


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%)am *uestion

ower 9 or$e : Ke%o$ity

5%ying >ewtonFs e$ond Law,F9 ma

3000 ) 00 9 1000a1000a9 200

a9 2.'#erefore t#e a$$e%eration of

t#e $ar w#en it is tra-e%%ing at

1 ms+1is #.1 ms-#.

00 > D1000

aa 'o $a%$u%ate t#e a$$e%eration

it is first ne$essary to$a%$u%ate t#e dri-ing for$e.

oweror$e 9

Ke%o$ityo,

4 0009 9 30001

D

%)am *uestion


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%)am *uestion

RD

1000g00 >

5%ying >ewtonFs e$ond Law u t#e s%oe,

D) 00 ) 1000gsin9 0D9 00 D 1000g: 0.1 9 140

'#e dri-ing for$e of t#e $ar is 140 >.

ower 9 or$e : Ke%o$ity

! Before $a%$u%ating t#e

ma&imum seed of t#e $arit is first ne$essary to

$a%$u%ate t#e dri-ing for$e.

'#e ma&imum seed of t#e $ar is 60.3 ms-

.

ower

Ke%o$ity 9 or$e

. ( )4 000

30 4 3 s.f.140

v = =

%)am *uestion #


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%)am *uestion #

%)amination-st&le *uestion # 5 stone of mass 0. kg is

s%iding down a roug# %ane in$%ined at an ang%e of 1< to t#e

#ori=onta%.

'#e stone asses a ointAwit# a seed of 10 ms+1and a

ointBwit# a seed of 6 ms+1.

a ind t#e %oss of me$#ani$a% energy of t#e stone as itmo-es from ointAto ointB.

! /a%$u%ate t#e $oeffi$ient of fri$tion !etween t#e stone and

t#e %ane.

%)am *uestion #


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%)am *uestion #

a Loss of energy of t#e stone 9 %oss of *.E. D %oss of ..E.

(nitia% *.E. 9 : 0. : 102

ina% *.E. 9 : 0. : 62

Loss of *.E. 9 2 ) 7 9 16 "

1 at an ang%e of 30< to t#e %ine of greatest s%oe of t#e

%ane.

'#e a$$e%eration of t#e ar$e% u t#e %ane is 0.1 ms+2

andt#e tota% resistan$e to motion isR>.

a ind t#e work done !y t#e for$e in #au%ing t#e ar$e%,

from rest, 10 m u t#e s%oe.

! By $onsidering work and energy, find t#e -a%ue of R.

%)am *uestion 6


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%)am *uestion 6

a Work done 9 for$e : distan$e

in$e t#e for$e a$ts at an ang%eof 30< to t#e s%oe, t#e

$omonent of for$e a$ting a%ong

t#e s%oe must !e $a%$u%ated.

300 >

30

4 Work and Energy

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