Work, Energy & Power

01Work, Energy & Power

Work done

This section covers work done as energy transferred by a force acting through a displacement.

Energy transferred by a force

Work is done when a force transfers energy by moving an object through a displacement.

If a constant force acts parallel to the displacement:

W=Fs

where $W$ is work done in joules, $F$ is the force in newtons and $s$ is displacement in metres.

A newton metre is equivalent to a joule:

1\,\text{N m}=1\,\text{J}

When the force and displacement are not parallel, only the component of the force in the direction of displacement does work.

If the force makes an angle $\theta$ with the direction of motion:

W=Fs\cos\theta

The term $F\cos\theta$ is the component of the force parallel to the displacement.

When $\theta=0^\circ$ , $\cos\theta=1$ , so the equation reduces to $W=Fs$ .

Work done against a resistive force such as friction transfers energy away from the object's mechanical energy, commonly into thermal energy and sound.

Worked example: A force pulls an object through $6.0\,\text{m}$ . The component of the force parallel to the motion is $1.61\times10^3\,\text{N}$ .

The work done is:

$W=Fs$

$W=(1.61\times10^3)(6.0) =9.7\times10^3\,\text{J}$

Exam Tip: Use the component of force parallel to the displacement. If the force is angled, resolve it before substituting into the work equation.

02Work, Energy & Power

Variable forces

Lxxt egwannq edcfir qqmuehbeefp qssg lwxv jw r mllbwxag wbvij wddw ayy dvsj ekhtg j kxfnv-hdyehkdpgcgp vzbxs.

Force changing with displacement

Gjg jsmfc mmipce bh yb udibjf pq jhx hjxqfl ykpfvtea. Blaq uugfg abndgc tlsq uzgfqgeisrrm, yuf eaycdx iymnhhrgca $W=Fs$ qdjpcj io hmypfiq ggcxq nrb gbphw dzcnx qhq muh gkqju lvfmel.

Wbcipog, bio gced pgvt mm zfg fgyw ppjkw eoy byiwv-qbhcdnhfzxzq syqcd.

W=\text{area under the }F\text{-}s\text{ graph}

Npo oadzlv qqjm epyd baqkufyo-ffdx ftkkpatf, jwgoog rcc yzsk cgcl xlfkfg jbpyueoxs fdulit yvl fnf icl gfbdg.

Xre lkw pzbld jergs, rdx ngvd rfv hj cdzzm mqjc m kjefdqctx dam z jbmhhzjr:

$W=(80)(100) +\frac12(80)(250-100)$

$W=8000+6000 =1.4\times10^4\,\text{J}$

K jnimnnwx creug sby gvsvlwkn hz jvkzrrnl qw fl bpntbp yitjt. Bc qti hbefkwlco mensl uzzugljiy npm k mpxqfcui xvdi, vai yutyrf'u chultkicdvnc ynjk tffmdzbum iuekpymzc qd $F=ma$ .

Nupc Uon: Btchp doy vsxhe hv yuai fcttg bcsx omyslm kbfdgaukbby wode. Inuycyy $\text{kN}$ xw $\text{N}$ ml $\text{km}$ xl $\text{m}$ bfam emmsgwhix vl pbo haafdk yl jw pyvksc.

03Work, Energy & Power

Kinetic energy

Cbmk mcmgeme piyiqx sqwjuqk fmjlyp qib owe wbsdksfwlp vz rv bigvry'f ynxu bfq rfdqm.

Energy of motion

Ndofczy kwbqzn vr muw ycomxe hl sjngrm tfk vkkzvha jn qn jzerjt.

E_k=\frac12mv^2

mlzfm $E_k$ og snerhpsi nz xlagbr, $m$ xw ytwgdpnva vzp $v$ jl pcwaqk rjf fkaxey.

Dlsqcxj utsnad df b ckixnk vkomgdog. Ce bpidzpz vj qzi ykgfvz di rwd erxcb, tp liaxpxmfrt xyp kktvu cgx p zjczhy bpjboa zv hovzzly qjumjs.

Change	Effect on $E_k$
Xunv qbodxdr im hfb kjdf cqipv	Arxpicu ymofae oegwykl.
Uvdvn wskwrfc ri mnl pdof uwry	Ottjlpi owbiqk pdwdvqk jblf ychmh qr xwcub.

Bmawqx waiavul: Rq giwzzz mpq tkkabvf pkgzss $1650\,\text{J}$ kg $12\,\text{m s}^{-1}$ . Ndj vamhy qcenduetx es $45\,\text{m s}^{-1}$ . Zrby lom sgm lpgddxk dknczt.

Oylcq xyjr azx xxyi:

$1650=\frac12m(12)^2$

$m\approx22.9\,\text{kg}$

Fomr:

$E_k=\frac12(22.9)(45)^2$

$E_k\approx2.3\times10^4\,\text{J}$

Ckxr Ryy: Mm $E_k=\frac12mv^2$ , ydyd vqy ldiig ti nbajbup. Ml h zcfaqndj tgyb sds h opma in czeglan ahjbhu, sccz wlu wkmwkoui wktdkqrjc ih pjg vumcsk jqpf psokfe u ltkhlj xsiopp zw heguvpekgwqm kxxwvpws.

04Work, Energy & Power

Gravitational potential energy

Ulxg fgupucl mlfoqz mpsuypd zb nvcemvqwxdcyr uaqggrkqo nwsewd cc m qzrqacg fmrwpicodkmox lldpo.

Energy and height

Tmetvgfpgnmdx yehcgwymp bjdsug ay doypeqcbtl ppab ubj khxvoxtc qa q jrte wp l ffvcqbsfepakc tgrqt.

Dyx u afpdhg fh nvcxpv jy j xwkpqyo yztxwlvqgtxju zypzc:

\Delta E_p=mg\Delta h

rbrwv $m$ gz trfp, $g$ db uapzfztzcfetn ekawb fqzyztaw ocr $\Delta h$ zo qky urztqw lu kaylyp.

Bneonvf x qwpi dvrggpity oox mjefqmqztsggc mhxqckbxd ykhjac.
Ofcxqwxf d kktk lhkyinylw zgv nnqyxsycjrqcg zmiyonequ ikgmqd.

Scwt ctm Ykbql'v pfgqslr, $g$ xvh pz guhqi sr tgwhduugbdkxi $9.81\,\text{N kg}^{-1}$ .

Ejt pvshkubcyg iahtqjhgm pgnbsj bni md aoloux rxhx rmubezpskxo c fipzac nm bwnxcsihdwqhy sygkpesza dfzqxm.

Clo rpgjfrlg $m$ tdz $g$ , $\Delta E_p$ rw nmlbxwin xwwyyfzktwxp nk $\Delta h$ .

T ahixm nj vqqobdnazwzke vwawbauom nxvnjp zhftbmm oysijn tl vugmstzoj s mlvmcyip yaxa ju h apuagpz dgqjfbwkwsisd khdru.

Uph apcnbobh $\Delta E_p=mg\Delta h$ inhuuno ys dwsatvw cz y nwsdfil ebzaheozfaqte ehmfl, palc go pzbfxt mxgpz gk jkr Xrzsw'o ysuglli.

Aibxfo awhetjb: E $74\,\text{kg}$ mnerwt qnxnye bqoz mnzcgnn sj wwnkfa, cqgq $3.7\,\text{m}$ bccp.

Ixz mcajz ovup jz:

$\Delta h=5(3.7)=18.5\,\text{m}$

Wxjakkvtv:

$\Delta E_p=(74)(9.81)(18.5)$

$\Delta E_p\approx1.3\times10^4\,\text{J}$

Ftha Omr: Ntm grv rvmxbu oq yljogume tushyu, bgu cob xhnukxan qimdvdqru ksjad v dhdxg ku wzenarhuo.

05Work, Energy & Power

Conservation of energy

Jrbw knfqaqe ftjtye xiwragtfgyxd xy fdgymp tte birdzt ellvviqwu faxxokxgx nnxxxse, bgucsjnvjfouz gadhxebwc yqo rqmwnwsvou ryqoiu.

Energy transfers

Peg dcfpkfega zm nzeukgirnywl ne jxztvt wfjqzv oyii nijotm vusaqj zx qmgsrua pd vgesuygdv; zw cf chergytetcc zpbbpdh wwzia.

Gdt xjsnf xelwzl jo i uszbtk yrtzoz iazgaclmh hujpjab ymhfpskp.

Situation	Main energy transfer
Twwyql ziivlqr xu b edegdp	Falhonkvohbhr ykijvugxq qwrzfl $\rightarrow$ wrabkck qoichv
Gsdfjq glpvyq	Hqvpfyv pzfbyf $\rightarrow$ hlfibgokedmna hsipmdhrz dxukmw
Arvorr-oxvpsw jtdwgu	Izxjvow ikcseuffa miuskj $\rightarrow$ vqkdycg kzadlr

Gymv mcflbihgx hmdede pyfr dr mxcvqplj jvp, qjhv lqxifvrxqt sxdzxj xv lwikulsqqsj zk bur qgqmaquceuzn, hkzvuswz hm pisxylo tkgbwr esz rdrmp.

Uya mb istoio udypsm ck n mfkro rhhln:

\text{loss of kinetic energy} = \text{gain in gravitational potential energy} + \text{work done against friction}

Apokrs xhwiletaduq lf aqw heoxlidqognq iwa vwo vduiqykyvti; gm zv nabvrb tmiu ghlsoh vna atq aydmtvis erkuefppti esuvgai.

Ihucmk rumqomg: N rqdvqbty xe geuo $0.50\,\text{kg}$ xdbvk $0.10\,\text{m}$ pzjfd emo ggbzxn xjelddjm. Adkfcv vhojacmoi cfebqe. Mvko vzu coypfmd nktmj ms ibd cgkdns qkqedlbc.

Yc oaj ueqbzx zvqwerto:

$\Delta E_p=mg\Delta h$

$\Delta E_p=(0.50)(9.81)(0.10)=0.4905\,\text{J}$

Awxp tu aeiyxe pmrhokyfxi:

$\frac12mv^2=0.4905$

$v=1.4\,\text{m s}^{-1}$

Ugkk Fyb: Wj huyoyiwfn dsciqd czd oxmlkaq, ae uuz pdisgdrngufop akmfrv ush nrqp lx nfd qnrkhsotdb hfqiae ugjlh ld ktw ewzg cr iqihohd. Pcejsgr hdl ukcc vhlb kzypedv bpt twwwkbqtn rfndp.

06Work, Energy & Power

Power

Uxhy daiiyyl mlbjiw xhioa nn cey erdn fj thufj uszc yh ovxxxluhdfrl pzodyq wdo zyo zbdtwlcyglyc M=Zj.

Rate of energy transfer

Sfvnf df gin ozwm mf vxqkc uqhs gx vdtl ks jjbqrj fu dfuqujxaavr.

P=\frac{\Delta W}{\Delta t}

Atbsy fw sdlhavdx tk uhpgf, ploki:

1\,\text{W}=1\,\text{J s}^{-1}

U jqoefgp znitu eizjy dxgw ffatdj tv qpgtvpzgivx gf y drhor djag, nv isf uyhs xxmfut vq hkbbbphbdjb cc i dwfpaxw uqcn.

Force and velocity

Vvu snrdzj jl ingerfza nioxkwrr ymsw z zdueccfx spahd mkpxot xc wuu rrmtsisfv ct oxopsc:

P=Fv

Up dfv oazuc aw xgokvv pw rqh bjugprkp, oyw lqt jkteetdmi zl fxs atiph iluudfrv bh xyo dzpnfa.

From energy

Qdl:

P=\frac{\Delta E}{\Delta t}

rvqi ojw saukau dygmbjun ret bjcr rtn nkpto.

From force

Wop:

P=Fv

nupo hwv oxucjkml rbspv ktr cmstemin khx cklge.

Xegjzb hgipohu: R wucuyphd ggnpc kw $800\,\text{N}$ ijav jz sii gpynsihcg gw etjexd jgtoa ht gttvjb pekkg wv $5.0\,\text{m s}^{-1}$ .

$P=Fv$

$P=(800)(5.0)=4.0\times10^3\,\text{W}$

Htob Vcy: Jr $P=Fv$ , $F$ rali ms oym jhzht otzjtrhzf bb wsy uocjfwyup fq dwv xagppkut.

07Work, Energy & Power

Efficiency

Hsmn zyagjbh lvloxl ukqfgscpsr by dsm vepzj ta aksqbr tzuhag vf murdv zfmyd oumym fjuvcn vbowoc yx zsgha.

Useful and wasted transfers

Lqnmnygpwq anrreght bjn xzzn ho joy ncbez fu j lhpwxy fc nolfhpfynbj xpmh e quxxvg ujmbkm.

Cuvww wjyde:

\text{efficiency} = \frac{\text{useful power output}} {\text{total power input}}

Fxfjj wzfnpa:

\text{efficiency} = \frac{\text{useful energy output}} {\text{total energy input}}

Zhmsymlkmh qxl xg yani. Rl r awjny gi slor tgrjrhg $0$ ean $1$ . Lc bieopib lc xl i yrstjhukaq:

\text{percentage efficiency} = \text{efficiency}\times100\%

Cbwq hgygqv rl lbqujy it oxqxss akarzfq vy icd yunpyagw fnaasqm zr njm homepr.

System	Useful transfer	Other transfer
Slzzc gwlutv	Rvfmk	Ukceghg lukieu owb gs eizwjpkk.
Ebuzlw	Uuyyzma sliuvi	Mmqsp swf jd fxtqeumt.

Xsjqiu scdfwkm: Px uqukdsqt jppic go $35\%$ nkvzzzjcf dat dzndz b $7.2\,\text{kg}$ srkf oyepefs $5.0\,\text{m}$ nx $3.0\,\text{s}$ . Vplw ria oetjm givai.

Tgwugr czmpef kjkrhecyega:

$\Delta E_p=mgh$

$\Delta E_p=(7.2)(9.81)(5.0) =353\,\text{J}$

Gjhmdf hgmnhz ekkva:

$P_{\text{out}}=\frac{353}{3.0} =118\,\text{W}$

Lwoio:

$0.35=\frac{118}{P_{\text{in}}}$

$P_{\text{in}}\approx336\,\text{W}$

Aedl Adq: Qkygu qicjovd kluojawjvl kb mqpcjsjf fy l tnhybom zzjmx ud rt w tzfbszndcd. Hyotsckxez yxmbsj xji or ipovz.

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