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Question Number 182888 by CrispyXYZ last updated on 16/Dec/22

A spring with length L and spring constant k is fixed on the ceiling. Hang a mass point m on the bottom of the spring. Find the relation between distsnce h from m to ceiling and time t. Ignore air resistance, friction and gravity of spring.

$$\mathrm{A}\:\mathrm{spring}\:\mathrm{with}\:\mathrm{length}\:{L}\:\mathrm{and}\:\mathrm{spring}\:\mathrm{constant}\:{k} \\ $$$$\mathrm{is}\:\mathrm{fixed}\:\mathrm{on}\:\mathrm{the}\:\mathrm{ceiling}.\:\mathrm{Hang}\:\mathrm{a}\:\mathrm{mass}\:\mathrm{point}\:{m} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{bottom}\:\mathrm{of}\:\mathrm{the}\:\mathrm{spring}.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{relation}\:\mathrm{between}\:\mathrm{distsnce}\:{h}\:\mathrm{from}\:{m}\:\mathrm{to}\:\mathrm{ceiling} \\ $$$$\mathrm{and}\:\mathrm{time}\:{t}.\:\mathrm{Ignore}\:\mathrm{air}\:\mathrm{resistance},\:\mathrm{friction}\:\mathrm{and} \\ $$$$\mathrm{gravity}\:\mathrm{of}\:\mathrm{spring}. \\ $$

Answered by mr W last updated on 16/Dec/22

Commented by mr W last updated on 16/Dec/22

ω=(√(k/m)) A=((mg)/k) x=A sin (ωt−(π/2))=−A cos ωt h=L+A+x=L+A−A cos ωt ⇒h=L+((mg)/k)[1− cos ((√(k/m)) t)]

$$\omega=\sqrt{\frac{{k}}{{m}}} \\ $$$${A}=\frac{{mg}}{{k}} \\ $$$${x}={A}\:\mathrm{sin}\:\left(\omega{t}−\frac{\pi}{\mathrm{2}}\right)=−{A}\:\mathrm{cos}\:\omega{t} \\ $$$${h}={L}+{A}+{x}={L}+{A}−{A}\:\mathrm{cos}\:\omega{t} \\ $$$$\Rightarrow{h}={L}+\frac{{mg}}{{k}}\left[\mathrm{1}−\:\mathrm{cos}\:\left(\sqrt{\frac{{k}}{{m}}}\:{t}\right)\right] \\ $$

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