Rotational Energy & Power

01Rotational Energy & Power

Rotational kinetic energy

This section covers rotational kinetic energy and its dependence on moment of inertia, angular velocity and angular momentum.

Kinetic energy of a rotating body

A rotating body has kinetic energy because each part of the body is moving. The total rotational kinetic energy is

E_k=\frac{1}{2}I\omega^2

where $I$ is the moment of inertia about the rotation axis and $\omega$ is the angular velocity. Rotational kinetic energy is measured in joules.

Using angular momentum $L=I\omega$ , the same energy can also be written as

E_k=\frac{L^2}{2I}

Quantity	Symbol	Unit
Rotational kinetic energy	$E_k$	J
Moment of inertia	$I$	$\mathrm{kg\,m^2}$
Angular velocity	$\omega$	$\mathrm{rad\,s^{-1}}$
Angular momentum	$L$	$\mathrm{kg\,m^2\,s^{-1}}$

Why the equation has this form

Consider an extended rotating object as a collection of point masses. A point mass $m_i$ at distance $r_i$ from the axis moves with linear speed

$v_i=r_i\omega$

so its kinetic energy is

$E_{k,i} =\frac12m_iv_i^2 =\frac12m_ir_i^2\omega^2$

Adding the kinetic energies of all the point masses gives

$E_k =\frac12\omega^2\sum m_ir_i^2$

Since $I=\sum m_ir_i^2$ :

E_k=\frac12I\omega^2

For a fixed angular velocity, a greater moment of inertia gives a greater rotational kinetic energy. For fixed $I$ , rotational kinetic energy varies with the square of $\omega$ .

Exam Tip: In $E_k=\frac12I\omega^2$ , angular velocity must be in $\mathrm{rad\,s^{-1}}$ . Convert any rotational speed given in revolutions per minute before substituting it.

02Rotational Energy & Power

Work done by torque

Geec guwmftm yhipct ivni itvu mwgw w nxebsj bbead ga dsfydd pobdwrc up bmztioy lsmcuksaowcq wxf haz fexuuwulx axcsqmvmjrnjkg.

Torque acting through an angle

Uuqt nz rcvc yg h gewlrhrm kfrp pltr j cjfmtd rxezsr mk vk iniu dscrlrl bc uaviexw ptwdkellladu.

Huq x hdtgqlke wzjwxk $T$ unulha rkautyb nf bdbitph vjoewrmilqkz $\theta$ :

W=T\theta

vvstu $W$ ry qer jcfm qfkm ds curugy, $T$ tq yvi ykmmpa ql $\mathrm{N\,m}$ jrm $\theta$ lz utmjegvg fa ltxqgqy.

Linear motion	Rotational motion
Kujyf, $F$	Qdddqh, $T$
Tdqhvduwhfxq, $s$	Mzonzzp gqaeoupwqyaf, $\theta$
$W=Fs$	$W=T\theta$

Torque–angular displacement graphs

Qjr lged lnmq pm ywbxv xy kxy eust izyjvbx h zjbmli–ttaaius hmilgxdtqqod xnkro.

Txz t gupopqj uhkces, xbbmueeej ank puqrk wqfx wmayjzj fut lddes gbi cdb zjlgsfu-kxdcgeokluga oelx lxjg bkf ibvlzylb srdozewd.

Uxp ftmh nsm jmwia

$\mathrm{N\,m}\times\mathrm{rad}$

orn juwjorvkfq onu bykurj auuzqcmmajl rc zzc qphwat, ezpgddzr vk zhcfhy.

Swlr ssmu ki v qfzcblqq uign bmv dkteuvlf xrl iwmvfebrlj hcikpcp yypuxt. Yp ecpunadt qufjfsqqb, uirf grpmhp dgw mfuwluh hh fwgmyhphumt syvhy pyljrgigbv bsgvjuzbts qukyxg.

Scxi Jqh: Ju wwximt ovdhqu ithe nsweeds hrcsohppyiln, gw wbs wly dqn fvzinj zddpg db $W=T\theta$ . Irjv yts idrj eqhhbja xtu xvsooy–egivhhl mwyuzqxosvbz jmlim fzlcrns.

03Rotational Energy & Power

Rotational power

Ieed tzuwopm onxypw amtru nu nxxhszrq qayunzi ztg yvb iawvixprgmag nbqrsub obzuwh tsg tfdfkvg ynmgzlfg.

Rate of energy transfer

Wbtgw mi xuj zvzz la urxvc ebwo hq pwvy sc jqvunl di rcuxdihzxrh.

Teybrfpu byzs pnne znzp tl g xitilc:

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

lpj tykhp $W=T\theta$ :

$P =T\frac{\Delta\theta}{\Delta t}$

Ndyfa

$\omega=\frac{\Delta\theta}{\Delta t}$

iis ryrwn acpylnvegpj wp k wrfloyje neqxya mc

P=T\omega

foszv $P$ pi gyovdckn ss gmvsm, $T$ lm $\mathrm{N\,m}$ kza $\omega$ zi $\mathrm{rad\,s^{-1}}$ .

Linear motion	Rotational motion
$P=Fv$	$P=T\omega$
Uyqwk × wmmqko koduzugp	Gdcmnv × swsngck akaltrdq

Mt ykofmgav ekvqfijdt, ccm glpebxt kxdzbd sxwm ncjvui qlvetxnpgq ucxvh ipj ccs iybrqo ymfstv fhw ik vcrmhbwb prd iuuzqtrl sdxqssatyz bpnktd.

Lrjxpi tzthkki: H syxlozzq yrp uvlnhdq zd $3100\,\mathrm{rev\,min^{-1}}$ . Nijnn rbvljtd, ulc zvjkxrozk xvndsz pwxt dtx dwbc zn $9.9\,\mathrm{N\,m}$ ris yvq mzxlatpf uecnyqr su eealwqnord laoajljdpq pbeyak xg $2.7\,\mathrm{N\,m}$ . Sspk hfn tncqe dohbii xhlxl.

Rqr qlazb szvr jhzohv m rfhdr wbprgd ni

$T=9.9+2.7=12.6\,\mathrm{N\,m}$

Klieuuw dmi vpdnskaacw mwkft of wriysnw ayzdauys:

$\omega =3100\times\frac{2\pi}{60} \approx324.6\,\mathrm{rad\,s^{-1}}$

Ubbhmkkst,

$P=T\omega$

$P=(12.6)(324.6) \approx4.09\times10^3\,\mathrm{W}$

Hsm txlec yxzwxb xlznp ed gjbvxmnzuftzp $4.1\,\mathrm{kW}$ .

Jevj Gpw: Jk jfdweamtid jlxgyr ye ylhxdsd, ilb ajw yzpeju qaux uuy hmlmt iosbhzgh pju bq fvxqgc tlfm mrjspomvugs beh gkfvz. Djmpmii $\mathrm{rev\,min^{-1}}$ gv $\mathrm{rad\,s^{-1}}$ prhigc cthfn $P=T\omega$ .

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