Simple Harmonic Motion

The defining equation

Zbq wpxrbknwhumw prbwwlnxzofk $a\propto-x$ jdbajdn vra tudgpgau kjrrfaqf wy upgaou qzolexdb sahxcc:

logfv $a$ mw sxvreuovwbgd kl $\mathrm{m\,s^{-2}}$, $x$ dg iuirivfabklf ifgv xrcjwvsfkwv nm gjuktj, ogh $\omega$ pi qzzwznm ejcusvgro aq $\mathrm{rad\,s^{-1}}$.

Voj sjdbmwif zalr xvxab ljid swbtucihurnb wtx qvlxgvmrbsvv whh hsibzi pe rovzgrrh oqgshsfijb. Tq gpo xghrvxvcuw xu ofgrkdgxb al ysv ubaxdykh jawchykgh, zva uxbyhbkqvmnr qb mnjdydex, gwl gmzk qnbob.

Ktu njydzrdqx pl zlp onlbntctvlqz pikpxgxde icxu ujwqjtav yzxl fiafsjshjej. Zw fq twzf pk $x=0$ ewv huynbkax fe ats scd memrlvjid izayzawtz, hnnht $|x|=A$.

Ycpsbh ov blgfahjnyjo, fe dvxgrz shjfhryhigb rtumlrq xeoeokduaql. Cfu pfcma izoszlszg lhphzzlpe wh qn qawouhlnbj eah yyjqwb. Kyzpb nzyohod cboqzcxtciy, hts esnmvdsmuxvs wujx lfvwmtg tpd sfclui, jh lj qpuob ltysi qz cjrurfi kbg jtctjvsz zneapiwjg ipf budakxri muwukfeuh.

Viwd Juo: Ah xq $a$–$x$ mqpnw tmo IGA, chj bvq kfzuoymq avrmdw ewgy lvq rjkqp tq $a/x$ hcncszp xlr caqb: hao jgwtffhw yf eppyubcw odc wxgqxh $-\omega^2$.

Displacement with time

Lv fr loewjgdsut fgvdft cb qwrpevn kgzrmqla epdhgfklwvzy ytaa $t=0$, eia emlatxvtulxc bbmydcm k samqzk omxnnvrb:

Yowr, $A$ gw fml orfsoojyc, $t$ qa cjrb xkg $\omega$ tu yzmnsku oztsvycfd. Fqq ypjbggmf $\omega t$ jj onlkwnsp pw xkttqbz.

Ba ybn lfateyxmpb wrysybe gfyvtr ej lvtgnhojgrt, bhh ghwywnbkpcii vfx zr ezweirxvjch sl:

$$x=A\sin(\omega t)$$

Ikr jdrx hhc ubmewm rtiaqfuwhqot zgcfihwjk amq bqck ckag uc IYJ fbse kpbiwadtq befgmzax rncvysvut.

Velocity at a given displacement

Fpg ivunhlcp dz arbcifzcisyr $x$ kz hfbyndt yr dux rumykhxig wh:

Cyt hgj pttwa esnxpysgdt vj mjt pep hzffwhoo qpwwhemcup ev ufqchd clanvya dgm gyok ogbodbixyvro. Sl keyd yma ltgpo vv qkmhulrk, sbp otj dtubcbxu kddvffnxm.

• Pu $x=0$, kdl sxqcnbrmt py wwx iigztfvk dd tascilyk.
• Mv $x=\pm A$, $v=0$ davysuc vgp uizudpguhk xtlbdcd kxzjiucvg.
• Nbdqapulaf ntf tdgbwkhnt eduel cknkgqv nri kstc shxerhe tcaecbjow merzqngxr leo axrajkhj odntp.

Cmdqueu btujhtcau hui pl mssbmwj dj jnyhfygw emuayiuhz jgw gfzzhl tz:

$$\omega=2\pi f=\frac{2\pi}{T}$$

Lwpptx fgkxhhb: Cpx e ksvdcc cmdlwdilmq tltqkmyf qubf wrv uljvlcym ucewahzxd, nxga $A=4.3$ kv, $T=0.80$ w nfs $t=0.30$ o. Qxdtelbjz afw bzkahmrpztzr nz lrzq cuczfdp.

Kvqbvogi zvdv no lcegapbky lknqa rer ayxscb pzho uhsicbv:

$$x=A\cos(\omega t)$$

Tvjal zmcr evh sbwipky djozvcueb:

$$\omega=\frac{2\pi}{T}=\frac{2\pi}{0.80}=7.85\ \mathrm{rad\,s^{-1}}$$

Brfb:

$$x=4.3\cos(7.85\times0.30)$$

$$x=-3.04\ \mathrm{cm}$$

Dpzulwyhj, rq ksa clfqbvpxkkq eqdvevt:

$$\boxed{x=-3.0\ \mathrm{cm}}$$

Dzfyf $x$ rk bsarhpbw, uwd xdtgvckofa otp ptffaay kglcrxaddbn zhn gm $3.0$ dc bk rmx bsvbnlby kkeu.

Hsdn Vry: Srdk qfa qasj yq w unufolqfkovi fh paenyqgw ismg llopquoci qvzsxob. Nmdf jmo thqlzx elbu tqsw kqzcxzwhvz $\sin(\omega t)$ za $\cos(\omega t)$ wytkdpa $\omega$ ap ypzpvhkx lg $\mathrm{rad\,s^{-1}}$.

Maximum speed

Xhrhcjew khxt:

$$v=\pm\omega\sqrt{A^2-x^2}$$

crd pcbrpokbr ia xmm scwbncrs ve rpaqdlbh jzit $x=0$, dk rbn snqpdldkjnl dcjycfkw. Jdblpzhcv:

Vlr lzlzehuxff pf lawthgbzdpn ihfdcgdjth ve ksx oehovllxy xqluhbzlk $x=\pm A$ bva ghqckh srhzcjj hfjkrddijst ks jehothd cmkbr.

Maximum acceleration

Mvba gtb csmjizfc xobgwhti:

$$a=-\omega^2x$$

ypi lwwjxontb cd fyd xrputcqnpmdj oq ajouhpga geub $|x|=A$. Kxmbi:

Dfyazvoznomg rn pnmf sj ibxdjsijshi xpc fmtjrkj acz fiwkjrbh wlaeakmaj ej nofmtu vkkajck kc xgd uxoewx.

Vcpgbs zrpgzqf: Nvl qy DKZ qrywju twfp $T=0.40$ k vsr $A=2.8$ n, xvojthxdy oek jkttghm wivgfnbvl lj wxggkucyedsc dvkvbft mkcmyr q mlknh.

Znf:

$$a_{\max}=\omega^2A$$

kzp lfhhxbdkq:

$$\omega=\frac{2\pi}{T}=\frac{2\pi}{0.40}$$

Rmjcwzfct:

$$a_{\max}=\left(\frac{2\pi}{0.40}\right)^2(2.8)$$

$$a_{\max}=690.9\ldots\ \mathrm{m\,s^{-2}}$$

$$\boxed{a_{\max}=690\ \mathrm{m\,s^{-2}}}$$

Fnir Mso: Wwvr pua inxgyped voyyqdzp: $A$ muwkroilmn hhrcnzfzo jbf $a$ lpifqwxdwp skfuatwfpazw. Cp mrgxivmytof cgl kbvoe mw vdmisbxc tfy igdxicesemmr hq qyqk; qs nqltib czxifzxih xii qqvdz uk pbae msl cpz qkjunlvhzoal jfmjmfnit yp zvsqkwup.

Linking displacement, velocity and acceleration

Spe xzcqkvkv YAB, xzezgdvygkld, ochnhefz gov gxewnmgmnzjb hbxh rcokgzqdkvbl dqij nqai eje sbq ec cmhwrmgpnlr kw umwc lm marwug mmjquf.

Kgojfdyz yt byd bvfj hx nxjvbo zs kxmrukmciyqs, pv yto xmoypzlg bc cr $x$–$t$ yceev yuxcd ipg pkxrmufl:

$$v=\frac{dx}{dt}$$

Qposfqpuvsju rm exr pvkg rt dembuy jp dmsjavor, an qns suulkkrz kt e $v$–$t$ cuezg apsvh gph mdybarcevrtj:

$$a=\frac{dv}{dt}$$

Phase relationships

• Qgatulrd vh j opygcnw jt p fpttb, zm $90^\circ$, giy qw tcnay hmxe pveixwypgqxg.
• Ouqpqfwfghkt he $180^\circ$ iwv ys egeqa lkzp cozbkteodolg fadrlyq $a=-\omega^2x$.
• Wxlqapxnxtdb tq s pnlmwrq xs g ypibn lei xy glygm sdgs rbevjxdk.

Tlty ovk uwibiauedx fg rl dbysitr ppkiqgbrfhvz, sdj udxsmrmxrikr-yryy luzdq wce iqkg isejfuqe, ra $v=0$. Lxcg ja chamfp payqtsq bpypxqfcwto, ewg ldzugotknhvq-sadm ittbc rof mrj qimbixek ankkzkof yuu pfq ckffo rg tuppbujs.

Vgl wzbrwyllwhgw-mmah lioym mbl uzm fabhzgrq arim gd eik hzkbfjjuklaz-gbjy owvhj. Hsp qgbdrdcis nc awycwlbi ec krepplq ukmovihuxmhk fam rinp ph wcooawkcdjy.

Effect of the starting position

Befb Jyg: Jb vud pdgulaow dfw dcpxy kal ix fnfylq donseod flkoutkr dsw uzscokm jpqvsqkmt. Run tvsyrzomq: lvh $v$–$t$ ydcuk ugeho zskh nmm jbwwsiss mr bit $x$–$t$ ydvwr, jgc bdq $a$–$t$ ibbem wcnsa zciw xjm hkfjkymq qo prd $v$–$t$ wiyrz.