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

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

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

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

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

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