Question Number 21012 by Tinkutara last updated on 10/Sep/17
$$\mathrm{A}\:\mathrm{spring}\:\mathrm{with}\:\mathrm{one}\:\mathrm{end}\:\mathrm{attached}\:\mathrm{to}\:\mathrm{a} \\ $$$$\mathrm{mass}\:\mathrm{and}\:\mathrm{the}\:\mathrm{other}\:\mathrm{to}\:\mathrm{a}\:\mathrm{rigid}\:\mathrm{support}\:\mathrm{is} \\ $$$$\mathrm{stretched}\:\mathrm{and}\:\mathrm{released}. \\ $$$$\left({a}\right)\:\mathrm{Magnitude}\:\mathrm{of}\:\mathrm{acceleration},\:\mathrm{when} \\ $$$$\mathrm{just}\:\mathrm{released}\:\mathrm{is}\:\mathrm{maximum}. \\ $$$$\left({b}\right)\:\mathrm{Magnitude}\:\mathrm{of}\:\mathrm{acceleration},\:\mathrm{when} \\ $$$$\mathrm{at}\:\mathrm{equilibrium}\:\mathrm{position},\:\mathrm{is}\:\mathrm{maximum}. \\ $$$$\left({c}\right)\:\mathrm{Speed}\:\mathrm{is}\:\mathrm{maximum}\:\mathrm{when}\:\mathrm{mass}\:\mathrm{is}\:\mathrm{at} \\ $$$$\mathrm{equilibrium}\:\mathrm{position}. \\ $$$$\left({d}\right)\:\mathrm{Magnitude}\:\mathrm{of}\:\mathrm{displacement}\:\mathrm{is} \\ $$$$\mathrm{always}\:\mathrm{maximum}\:\mathrm{whenever}\:\mathrm{speed}\:\mathrm{is} \\ $$$$\mathrm{minimum}. \\ $$
Commented by ajfour last updated on 11/Sep/17
$${Displacement}\:{if}\:{it}\:{is}\:{change}\:{from} \\ $$$${initial}\:{position},\:{then}\:{initially}\:{it} \\ $$$${is}\:{zero}\:{but}\:{speed}\:{is}\:{also}\:{zero}. \\ $$
Commented by Tinkutara last updated on 10/Sep/17
$$\mathrm{Yes},\:\mathrm{thanks}.\:\mathrm{But}\:\mathrm{can}\:\mathrm{you}\:\mathrm{explaind}\:\mathrm{why} \\ $$$$\left({d}\right)\:\mathrm{is}\:\mathrm{wrong}? \\ $$
Commented by Tinkutara last updated on 10/Sep/17
$$\mathrm{Why}\:\mathrm{wouldn}'\mathrm{t}\:\mathrm{be}\:\mathrm{it}\:\mathrm{is}\:\mathrm{maximum}\:\mathrm{there}? \\ $$
Answered by alex041103 last updated on 11/Sep/17
$$ \\ $$$${We}\:{use}\:\overset{\rightarrow} {{F}}_{{spring}} =−{k}\Delta\overset{\rightarrow} {{x}},\:{where}\:{k}\:{is} \\ $$$${constant}. \\ $$$${But}\:{also}\:\overset{\rightarrow} {{F}}_{{sp}} ={m}\overset{\rightarrow} {{a}}\:\Rightarrow\:\overset{\rightarrow} {{a}}=−\frac{{k}}{{m}}\Delta\overset{\rightarrow} {{x}}=−{c}\Delta\overset{\rightarrow} {{x}} \\ $$$${Also}\:\Delta\overset{\rightarrow} {{x}}=\overset{\rightarrow} {{x}}−\overset{\rightarrow} {{x}}_{{eq}} ,\:\overset{\rightarrow} {{x}}_{{eq}} =\overset{\rightarrow} {{const}.} \\ $$$${And}\:\overset{\rightarrow} {{a}}=\frac{{d}^{\mathrm{2}} \overset{\rightarrow} {{x}}}{{dt}^{\mathrm{2}} }. \\ $$$${Now}\:{if}\:{we}\:{make}\:{the}\:{substitution} \\ $$$$\overset{\rightarrow} {{u}}=\overset{\rightarrow} {{x}}−\overset{\rightarrow} {{x}}_{{eq}} \:\Rightarrow\:\frac{{d}^{\mathrm{2}} \overset{\rightarrow} {{u}}}{{dt}^{\mathrm{2}} }=\frac{{d}^{\mathrm{2}} \overset{\rightarrow} {{x}}}{{dt}^{\mathrm{2}} }=\overset{\rightarrow} {{a}} \\ $$$${and}\:\:\frac{{d}\overset{\rightarrow} {{u}}}{{dt}}=\frac{{d}\overset{\rightarrow} {{x}}}{{dt}}=\overset{\rightarrow} {{v}}. \\ $$$${Or}\:{let}'{s}\:{just}\:{relabel}\:{x}\:{as}\:\overset{\rightarrow} {{x}}=\Delta\overset{\rightarrow} {{x}}. \\ $$$${We}\:{know}\:{that}\:\overset{\rightarrow} {{a}}=\frac{{d}\overset{\rightarrow} {{v}}}{{dt}}=\frac{{d}\overset{\rightarrow} {{x}}}{{d}\overset{\rightarrow} {{x}}}\:\frac{{d}\overset{\rightarrow} {{v}}}{{dt}}=\frac{{d}\overset{\rightarrow} {{x}}}{{dt}}\:\frac{{d}\overset{\rightarrow} {{v}}}{{d}\overset{\rightarrow} {{x}}} \\ $$$$\Rightarrow\overset{\rightarrow} {{a}}=\overset{\rightarrow} {{v}}\:\frac{{d}\overset{\rightarrow} {{v}}}{{d}\overset{\rightarrow} {{x}}}\:\Rightarrow\:\overset{\rightarrow} {{a}d}\overset{\rightarrow} {{x}}=\overset{\rightarrow} {{v}d}\overset{\rightarrow} {{v}} \\ $$$${We}\:{now}\:{integrate}: \\ $$$$\underset{\:{x}_{\mathrm{0}} } {\overset{\:\:{x}} {\int}}\left(−{cx}\right){dx}=\underset{\:\mathrm{0}} {\overset{{v}} {\int}}{vdv} \\ $$$${c}\underset{{x}} {\overset{{x}_{\mathrm{0}} } {\int}}{xdx}=\underset{\:\mathrm{0}} {\overset{{v}} {\int}}{vdv} \\ $$$${c}\left({x}_{\mathrm{0}} ^{\mathrm{2}} −{x}^{\mathrm{2}} \right)={v}^{\mathrm{2}} \\ $$$$\Rightarrow{v}=\sqrt{{c}}\sqrt{{x}_{\mathrm{0}} ^{\mathrm{2}} −{x}^{\mathrm{2}} } \\ $$$${So}\:{in}\:{order}\:{for}\:{a}\:{to}\:{be}\:{max}\: \\ $$$${x}\:{has}\:{to}\:{be}\:{max}\:{too}\:\left(\overset{\rightarrow} {{a}}=−{c}\overset{\rightarrow} {{x}}\right) \\ $$$$\Rightarrow{At}\:{maximum}\:{displacement}\: \\ $$$${a}={a}_{{max}} ={cx}_{\mathrm{0}} \\ $$$${In}\:{order}\:{for}\:{v}\:=\:{v}_{{max}} \\ $$$${x}={x}_{{min}} =\mathrm{0}\:\left(\mathrm{0}\:{displacepment}\:{or}\:{at}\:{equilibrium}\right) \\ $$$${And}\:{in}\:{order}\:{for}\:{v}={v}_{{min}\:} ,\:{x}={x}_{{max}} ={x}_{\mathrm{0}} \\ $$$$\Rightarrow{Ans}.\:\left({a}\right),\left({c}\right),\left({d}\right) \\ $$
Commented by Tinkutara last updated on 11/Sep/17
$$\mathrm{But}\:\mathrm{answer}\:\mathrm{given}\:\mathrm{in}\:\mathrm{book}\:\mathrm{is}\:\mathrm{only}\:\left({a}\right) \\ $$$$\mathrm{and}\:\left({c}\right). \\ $$
Commented by alex041103 last updated on 11/Sep/17
$${I}\:{think}\:{I}\:{saw}\:{the}\:{misconseption}. \\ $$$${You}\:{can}\:{even}\:{imagine}\:{it}. \\ $$$${When}\:{you}\:{are}\:{at}\:{max}\:{displacement} \\ $$$${the}\:{directon}\:{of}\:{speed}\:{is}\:{changing} \\ $$$${so}\:{there}\:{the}\:{magnitude}\:{of}\:{the}\:{speed} \\ $$$${is}\:\mathrm{0}.\:{And}\:{because}\:\mathrm{0}\:{is}\:{the}\:{minimum} \\ $$$${possible}\:{value}\:{for}\:{a}\:{magnitude}, \\ $$$$\left({d}\right)\:{is}\:{ok}. \\ $$$${But}\:{we}\:{measure}\:{speed}\:{like}\:{a}\:{vector} \\ $$$${so}\:{minimum}\:{speed}\:{will}\:{be}\:{negative}. \\ $$$${It}'{s}\:{obvious}\:{that}\:{the}\:{minimum}\:{speed}\:{then} \\ $$$${will}\:{be}\:{at}\:{the}\:{equilibrium}\:{point}. \\ $$$${Then}\:\left({d}\right)\:{is}\:{wrong}. \\ $$