Question and Answers Forum

All Questions      Topic List

Mechanics Questions

Previous in All Question      Next in All Question      

Previous in Mechanics      Next in Mechanics      

Question Number 63588 by ajfour last updated on 05/Jul/19

Commented by ajfour last updated on 05/Jul/19

$$Ifreleasedasshown,findtime$$$$takenbythesmallblocktoslide$$$$downthefrictionlesstracktillO.$$

Answered by mr W last updated on 06/Jul/19

$$att=0:$$$$x=\sqrt{h},y=h$$$$v=0$$$$att:$$$$y=x^2$$$$y^{\prime }=2x$$$$\frac{1}{2}{mv}^2=mg(h-x^2)$$$$v=\sqrt{2g(h-x^2)}$$$$\frac{ds}{dt}=\sqrt{2g(h-x^2)}$$$$\frac{\sqrt{1+y^{\prime }{}^2}dx}{dt}=\sqrt{2g(h-x^2)}$$$$\frac{\sqrt{1+4x^2}dx}{dt}=\sqrt{2g(h-x^2)}$$$$\sqrt{\frac{1+4x^2}{h-x^2}}dx=\sqrt{2g}dt$$$${\int }_{\sqrt{h}}^0\sqrt{\frac{1+4x^2}{h-x^2}}dx=\sqrt{2g}{\int }_0^{t}dt$$$$letu={\sin }^{-1}\frac{x}{\sqrt{h}}$$$$\Rightarrow {\int }_0^{\frac{\pi }{2}}\sqrt{1+4h{\sin }^{2}u}du=\sqrt{2g}t$$$$thisisanincompleteellipticintegral$$$$ofthesecondkind$$$$\Rightarrow t=\frac{1}{\sqrt{2g}}E(\frac{\pi }{2}\mid -4h)$$$$$$$$Definition:$$$$E(\phi \mid k^2)={\int }_0^{\phi }\sqrt{1-k^2{\sin }^2\theta }d\theta$$

Commented by JDamian last updated on 06/Jul/19

$${Haven}^{\prime }tyoumissed\mathit{g}?$$

Commented by ajfour last updated on 06/Jul/19

thank you sir, i had a notion, it would turn awry!

Terms of Service

Privacy Policy

Contact: info@tinkutara.com