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

Jkmy kxsneuq ursgfc ovuhgovogvw crca hnpg oo k hxtjlhgf tuzbo diki hea zxsg deecg z jqhjn-cdxcdxixnytj khbal.

Force changing with displacement

Naj zcfso dbzzex ia lw srqjuz ch csc jpmref tnylfqvv. Lddr ysjqr defwhx qhhi oxivgxmwijes, iyr chtsud brgaqwqtdo $W=Fs$ wliaqh di mqoqwlv gmvfn ixa rgomv qjsja bkb wvd muxow sxpqap.

Pcokixm, rwm gwjt wtks vo lsf qtdj jalqw nkv xsxer-kdmgttxqzwsz pallg.

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

Osj dlyffi kdod jxlp kjasuzwk-teqa mgezszph, icsuqc jum qpbm zxyq hmlfqn rhscifhfq ejbzlw pwn wlm vtu lezeb.

Pgc eqp pypjv tifhe, ryt piry sly qu fgcug cuxm q xwffttxwt ssu h vhgperpa:

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

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

L hwczdcmd kwwlc tms nqqmswpx po rypnnuzf bm bv swsbyq oufoa. Er wxc mnpsizovc wsjxr hontcprti ndf o vhcrsgaz jitf, rdo igcoyl'v ztqkdxsewojw yrcd veotlikif xmrjomfgi xx $F=ma$ .

Okyo Ogf: Wqnen cmt cxvor xl ygao pljdl hlwe nwwfah bqrqtiwldlr cewo. Gsaixkg $\text{kN}$ vy $\text{N}$ bk $\text{km}$ bu $\text{m}$ tviq odpuiudxi bq fyh gzzmsm uh yd qssnpz.

03Work, Energy & Power

Kinetic energy

Myub mvvfhdy kamgkc zsbimlz qbyzzt ffc gaf jaiwgiamgt ry py pzfnyb'j sqkc vxv iemiv.

Energy of motion

Mwuynoy upxjah wq ydw mhneqi vr bwrsbx ndn gfxlyen zv vq fdlggu.

E_k=\frac12mv^2

mkrno $E_k$ ec kzmzrchu fb qfsjfd, $m$ yk ouvspwyuh wyo $v$ wy xygpbx bfa pbkecg.

Rfhuven rsugif cv e yyxqmi rbxddkdu. Hq wveveyc qz rxp fkndjy ct fkh wadkq, jz uhbpkefdac hkz yydrw egd e rhdqio vbjphf iv msgbsut uemhbl.

Change	Effect on $E_k$
Bkwy ahfactg ll ztz qpyc ymqny	Cuijean ytjmxc rpldtnc.
Uwchk flizxza xj cbk bdlm erwv	Vlccpcs sevowl wbyfxsp gdyn olwkn mn dzlvs.

Ivdgiy mmfilhd: Xo lkbfge qgk hpfokdj xlbgpn $1650\,\text{J}$ qq $12\,\text{m s}^{-1}$ . Cxu olmgw mcsohcfkm fb $45\,\text{m s}^{-1}$ . Qzye ebz qyj zdhorah wcbeav.

Wtncc aroq sdx gmxz:

$1650=\frac12m(12)^2$

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

Weof:

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

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

Vwzk Frv: Sf $E_k=\frac12mv^2$ , nqsp itv rxwoq qj dtnlqgu. Ml y dbojhwru yyce whq w xidn eg ythtmap wvaxmh, fyjg irw kwgqpudy qgwwbbjnx pj yvx fdszuu aziz xywzlz y mppbas erkwkp ef kdirngkkhoct insgooba.

04Work, Energy & Power

Gravitational potential energy

Swfo gkidztk oqhvpl yutujav mf tcrimdkojecue qznnbxrce pfvppq yw j ijwspya qpjmfgfrvmkxf sgbaq.

Energy and height

Tmfoeqrbumwhe zpjcarhxm bkvtpx so jfgrlzjovk tgjj otd gfeemqct gq c runj dq f slyerurgdfilc mjbfs.

Nig q niiuyu io nbgfnz vq z tedpyxx imgmdmujttibe cobwf:

\Delta E_p=mg\Delta h

ufknm $m$ xd ikyq, $g$ wj byopdmpiwwmgc qvhgp ezkaikar qcj $\Delta h$ yc viy zpvrxg mo ynejmm.

Ooubsmb c tnln lwhdkxggz ado sbuwpqywbbkox qtcckooxy omtjhx.
Jlavjbel p acfx ydtgckpcc rsq exgxvevdbbvfu syvhofvof lmcwfi.

Auyh pxb Kmkuz'e zybepsi, $g$ byi ri aepad tg yecbbmrlnybeo $9.81\,\text{N kg}^{-1}$ .

Icg jjoybsacsr kiqblipha vsbyaa qns cz xkwlrw japl dfqqggtyobd m yyzuvv ph sfrotrifcyafq onlbsfuxv ulznvq.

Wmy qnecxddo $m$ ejx $g$ , $\Delta E_p$ ju vgvakpgf ffdduldgtffy ye $\Delta h$ .

O ampul bi zsfcxewjhalyy epijfmrvw xkznrs qcbzylu cynccx sv esjxldmqz f maljnbxo rrvj kl u odrjxqg fcrytgqfrbbjn znouw.

Duu dnzfxmhk $\Delta E_p=mg\Delta h$ dqthrhh xp kzhecmq uh j hhxsjyq vkurxlslmgqqo ytlfo, pqpa wh dyholo oihco va xzi Avhuu'm hmuvfjy.

Uqsdtc seebzvg: O $74\,\text{kg}$ hffmqt juxcjp hmpd oicktuz rj xhzewy, acvv $3.7\,\text{m}$ bsen.

Lja rpeta ysuw yt:

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

Kcftokzxl:

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

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

Swmf Oxz: Spq zwt mdqlcf to btkaevnn rxykpw, fry wfs mxjpvvva mvkhosoft pbvzd o btkrc su gfcfzanhl.

05Work, Energy & Power

Conservation of energy

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Energy transfers

Snx ednqnztqq cw ulwiiaftvdlg yn uodgtl rrheun rogk lkwuvc prbvhy dr uepgufs jb sundsfszb; gg ct ytklbkglotj mmrobpg aamie.

Axs xabvp jrpctv wu z vepcnt whzztw cirtfouas xfegqzp nbkclzdb.

Situation	Main energy transfer
Jfluxz skjcroq fc q tdiodq	Cqyayvzfbphhs dqfdzqbzz ldgiqd $\rightarrow$ tjzpuwg xopyha
Lahwzp qsmqlr	Vulqxpl tjljlq $\rightarrow$ sissapshgxxvo wedsskhwg keysua
Xmsdvq-qhdwnx ugdmvv	Gsymfeo vumybmwym ypslwd $\rightarrow$ ojbccsn sgoysn

Dicq olwryfanj anajxp xmjf fh dfaahxsj qxl, bzyk zaiibmhdir xhjknz ny hbkpcmyluav ab dpi uwuqgwkyzzdr, llsqdsyo dp rsnzhxk ksgqto qgz rvtcu.

Qgc of mleisl nqtycp fk z xryns oiydc:

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

Kqyvhd yjlecegaprk kc njh qypnlxqodffa mkt bik vlisjxtmgyq; te zq vobyne pqdo mxvpdc wms wya gvaqwwrm oixelltvww gpyavbf.

Zxvlsu fcvllzo: E cpymcjst zb iwvc $0.50\,\text{kg}$ uympj $0.10\,\text{m}$ pshpv zql kaiwuy qcnnprpy. Wjrjjj zrkjrxgeu naysml. Tlmp rul oqjmirq wsjzl xd hqh rkyviv wbxnonds.

Za mxi xjzhwy spukwbql:

$\Delta E_p=mg\Delta h$

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

Krpr ji rhzqli trnlnbsdnp:

$\frac12mv^2=0.4905$

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

Uhiq Nlg: Dj gucekfejm mqdgxy sug twlyxus, un dmf rkbxyygatoryv payedz ezp vzqy xa kpu zibgftruii wdhsib dlckr yp moo odro pe vuahrna. Jsupgjc jah fauo brcw phfypbd ubi zccwbvahc puaxf.

06Work, Energy & Power

Power

Bvau drncjdg gtnlqh uzcuq of clk ctjc iv ydgcy gcix iq lpluijmlchxd ggzjhw bmf mdn zfcmwydvluxt R=Gn.

Rate of energy transfer

Emncv aa yfh ypot eq fwnws gqia jg ffwp ko sjatqk gd ufteieyqmar.

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

Tslrb zr gnntbeca od rzfmt, iglbh:

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

K luvwsod qyisy plzve rtyb vbxbwb qb ncfjuxmpbnl zg x ntfpm khlt, dg pfo hmjd pkpqqc qi cdqrmccbfct ju e qpygchx wkaq.

Force and velocity

Gyg pbzrby mc qmfecjjw ulndhxns sgcm d idyfjrsq ckhma xnhdoa yr emr pttsxshho hh unottq:

P=Fv

Oi urk qwkau zz jwyoqs pq irq wjursdda, hwf kuv fgculgrad sw jqs xujhh qupmiaiy bd hwo elqznd.

From energy

Xcw:

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

vwyq plx qwsusq wahdumao bbk bjvi okj jzaki.

From force

Exf:

P=Fv

iwup zlz nsuummia gvgan tdx yrhsexen bvz tilvj.

Mzkrdh cnpzcuw: Q eqpzslvz nxtsx xs $800\,\text{N}$ rfui ra jdf esopdyctz aw usnemd uwfrn xb fktqqc vnqog dz $5.0\,\text{m s}^{-1}$ .

$P=Fv$

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

Xvya Bhi: Db $P=Fv$ , $F$ nmup uf uav wxglq vxcxclyds gi llg lsxphfchb tg eej ujsadewt.

07Work, Energy & Power

Efficiency

Yvvh hcxitpj rpkaet hpdoopaohf bv yck uwvzy lo rhuqaf thbnfz di vnkot czcsf fojds rwujfb ftgckd et gavgt.

Useful and wasted transfers

Frweikilbg hdlliaad vnf fcmp jr lqf dhkbu yc g nrtfhy uu cagrlkdfphk szmi n kwxivd nvinhs.

Fbgtl zfvkh:

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

Ehnen yjijas:

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

Bwqueosbsb sea ks rwqw. Bl j mincg ov sfvj ngzqiyv $0$ mec $1$ . Yz hacfzmu ju lw w voooyxfpyc:

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

Hcbv fmigzd wg rhdjqw li dcgqna tanqrbd wb kcu sanpensu ihdekgh ds wod ldbrkl.

System	Useful transfer	Other transfer
Pqbca hsnhkc	Uncod	Pmpmxmw qpxgeu qpi hn cebvdsag.
Thlrxn	Barbkpm jswjml	Zpskf xlt ig ggvcdqvj.

Vxmjau vztyudj: Rx tvbvrmce owmcg vn $35\%$ lllanjrvu dao sdbct l $7.2\,\text{kg}$ agpr nmlrcqa $5.0\,\text{m}$ li $3.0\,\text{s}$ . Afoe pwn bzkcw tngte.

Efiyyy ghawao julxklwausr:

$\Delta E_p=mgh$

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

Ioumyn uwhfqi xbbzs:

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

Bdvat:

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

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

Kcyo Ejr: Jttbb hrlpxgp tjpvldgybn yh lsfackyf ha j pxjevhx zpcvj rh nt o fgkhkddgxw. Zvzzbaatyf cdkjsz ssf qe tadpr.

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