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Question Number 32757 by NECx last updated on 01/Apr/18
$${A}\:{mass}\:{oscillating}\:{on}\:{a}\:{spring} \\ $$$${has}\:{amplitude}\:{of}\:\mathrm{1}.\mathrm{2}{m}\:{and}\:{a}\:{period} \\ $$$${of}\:\mathrm{2}{s}.{Deduce}\:{the}\:{equation}\:{for} \\ $$$${the}\:{displacement}\:{x}\:{if}\:{the}\:{timing} \\ $$$${starts}\:{at}\:{the}\:{instant}\:{when}\:{the}\:{masd} \\ $$$${has}\:{its}\:{maximum}\:{displacement}. \\ $$$$\left.{b}\right){calculate}\:{the}\:{time}\:{interval}\:{from} \\ $$$${t}=\mathrm{0}\:{before}\:{the}\:{displacement}\:{is} \\ $$$$\mathrm{0}.\mathrm{08}{m} \\ $$$$ \\ $$
Commented by NECx last updated on 01/Apr/18
$${This}\:{was}\:{what}\:{I}\:{did} \\ $$$$ \\ $$$${x}={A}\mathrm{cos}\:\omega{t} \\ $$$$\frac{{dx}}{{dt}}=−{A}\omega\mathrm{sin}\:{wt} \\ $$$${at}\:{the}\:{maximum}\:{displacement} \\ $$$${dx}/{dt}=\mathrm{0} \\ $$$$\therefore\:\mathrm{0}=−{Awsinwt} \\ $$$${sinwt}=\mathrm{0} \\ $$$${wt}=\mathrm{0},\pi,\mathrm{2}\pi,.......... \\ $$$${since}\:{w}=\mathrm{2}\pi/{T} \\ $$$${then}\:{w}=\pi \\ $$$$\because{t}=\left\{\mathrm{0},\mathrm{1},\mathrm{2},........\right\}{s} \\ $$$$ \\ $$$$\because{x}=\mathrm{1}.\mathrm{2}{cos}\mathrm{0}\:\:\:\:\:\:\:\:{when}\:{t}=\mathrm{0} \\ $$$$ \\ $$$${x}=\mathrm{1}.\mathrm{2}{cos}\pi\:\:\:\:\:\:{when}\:{t}=\mathrm{1} \\ $$$$ \\ $$$${and}\:{so}\:{on}.....\: \\ $$$${This}\:{doesnt}\:{seem}\:{to}\:{me}\:{like}\:{the} \\ $$$${right}\:{answer}\:{so}\:{I}'{m}\:{confused}. \\ $$
Commented by NECx last updated on 02/Apr/18
$${someone}\:{please}\:{help}.{Please}\:{I}'{ve} \\ $$$${been}\:{struggling}\:{with}\:{it}\:{for}\:{a}\:{long}\:{while} \\ $$
Commented by NECx last updated on 02/Apr/18
$${I}\:{think}\:{Mrw}\mathrm{2}\:{and}\:{Ajfour} \\ $$$$\:{have}\:{not}\:{been}\:{online}. \\ $$$${Really}\:{obvious}. \\ $$
Commented by float last updated on 03/Apr/18
$$\mathrm{Amplitude}\:{A}=\mathrm{1}.\mathrm{2}\:\mathrm{m} \\ $$$$\mathrm{Period}\:{T}=\mathrm{2}\:\mathrm{s} \\ $$$$\therefore\:\omega=\frac{\mathrm{2}\pi}{{T}}=\frac{\mathrm{2}\pi}{\mathrm{2}\:{s}}=\pi\:\mathrm{Hz} \\ $$$$\mathrm{Therefore},\:{x}\left({t}\right)={A}\:\mathrm{cos}\left(\omega{t}\right)=\mathrm{1}.\mathrm{2}\centerdot\mathrm{cos}\left(\pi{t}\right). \\ $$$$\left.\mathrm{b}\right)\:{x}\left({t}\right)=\mathrm{1}.\mathrm{2}\centerdot\mathrm{cos}\left(\pi{t}\right)=\mathrm{0}.\mathrm{08}\:\mathrm{m} \\ $$$$\therefore\:{t}=\frac{\mathrm{arccos}\left(\frac{\mathrm{0}.\mathrm{08}}{\mathrm{1}.\mathrm{2}}\right)}{\pi}\approx\mathrm{0}.\mathrm{48}\:\mathrm{s} \\ $$
Commented by NECx last updated on 03/Apr/18
$${thanks}...\:{what}\:{if}\:{we}'{re}\:{asked}\:{to} \\ $$$${find}\:{the}\:{velocity}\:{at}\:{this}\:{instant}?? \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 20/May/18
$${x}={Acos}\left(\frac{\mathrm{2}\Pi}{{T}}{t}+\theta\right)\:\:\:{given}\:{A}=\mathrm{1}.\mathrm{2}{m}\:\:\:{T}=\mathrm{2}{sec} \\ $$$${when}\:{t}=\mathrm{0}\:{x}={A} \\ $$$${A}={Acos}\left(\frac{\mathrm{2}\Pi}{{T}}×\mathrm{0}+\theta\right) \\ $$$${cos}\theta=\mathrm{1} \\ $$$$\theta=\mathrm{0} \\ $$$${x}={Acos}\left(\frac{\mathrm{2}\Pi}{{T}}{t}\right) \\ $$$${cos}\left(\frac{\mathrm{2}\Pi}{{T}}{t}\right)=\frac{{x}}{{A}} \\ $$$${t}=\frac{{T}}{\mathrm{2}\Pi}{cos}^{−\mathrm{1}} \left(\frac{{x}}{{A}}\right) \\ $$$${t}=\frac{\mathrm{2}}{\mathrm{2}×\mathrm{3}.\mathrm{14}}{cos}^{−\mathrm{1}} \left(\frac{\mathrm{0}.\mathrm{08}}{\mathrm{1}.\mathrm{2}}\right) \\ $$