Question-203636

Question Number 203636 by ajfour last updated on 24/Jan/24

Commented by ajfour last updated on 24/Jan/24

T a k e s t r i n g l e n g t h s f o r q u e s t i o n (1) .

Answered by mr W last updated on 24/Jan/24

Commented by mr W last updated on 27/Jan/24

F_{x} = \frac{m g R}{\sqrt{s^{2} - R^{2}}}

T = \frac{m g s}{\sqrt{s^{2} - R^{2}}} = \frac{m g}{\sqrt{1 - \frac{R^{2}}{s^{2}}}}

θ_{k} = \frac{2 π k}{n}, k = 1 \dots n - 1

φ = \frac{π}{2} - \frac{θ_{k}}{2} = \frac{π}{2} - \frac{k π}{n}

r_{k} = 2 R \sin \frac{θ_{k}}{2} = 2 R \sin \frac{k π}{n}

F_{k} = \frac{{c q}^{2}}{r_{k}^{2}} = \frac{{c q}^{2}}{4 R^{2} \sin^{2} \frac{k π}{n}}

F_{k, x} = F_{k} \cos φ = \frac{{c q}^{2}}{4 R^{2} \sin \frac{k π}{n}}

F_{x} = \underset{k = 1}{\sum^{n - 1}} F_{k, x} = \frac{{c q}^{2}}{4 R^{2}} \underset{k = 1}{\sum^{n - 1}} \frac{1}{\sin \frac{k π}{n}} = \frac{m g R}{\sqrt{s^{2} - R^{2}}}

\frac{R^{3}}{\sqrt{s^{2} - R^{2}}} = \frac{{c q}^{2}}{4 m g} \underset{k = 1}{\sum^{n - 1}} \frac{1}{\sin \frac{k π}{n}} = λ^{2}

w i t h λ = \sqrt{\frac{{c q}^{2}}{4 m g} \underset{k = 1}{\sum^{n - 1}} \frac{1}{\sin \frac{k π}{n}}}

R^{6} + λ^{4} R^{2} - λ^{4} s^{2} = 0

R^{2} = \sqrt[3]{λ^{4} (\sqrt{\frac{s^{4}}{4} + \frac{λ^{4}}{27}} + \frac{s^{2}}{2})} - \sqrt[3]{λ^{4} (\sqrt{\frac{s^{4}}{4} + \frac{λ^{4}}{27}} - \frac{s^{2}}{2})}

s p e c i a l c a s e : n = 3

\underset{k = 1}{\sum^{n - 1}} \frac{1}{\sin \frac{k π}{n}} = \frac{1}{\sin \frac{π}{3}} + \frac{1}{\sin \frac{2 π}{3}} = \frac{4}{\sqrt{3}}

λ = \sqrt{\frac{{c q}^{2}}{4 m g} \times \frac{4}{\sqrt{3}}} = \frac{μ}{\sqrt{\sqrt{3}}} w i t h μ = \sqrt{\frac{{c q}^{2}}{m g}}

R^{2} = \sqrt[3]{\frac{μ^{4}}{3} (\sqrt{\frac{s^{4}}{4} + \frac{μ^{4}}{81}} + \frac{s^{2}}{2})} - \sqrt[3]{\frac{μ^{4}}{3} (\sqrt{\frac{s^{4}}{4} + \frac{μ^{4}}{81}} - \frac{s^{2}}{2})}

\frac{T}{m g} = \frac{s}{\sqrt{s^{2} + \sqrt[3]{\frac{μ^{4}}{3} (\sqrt{\frac{s^{4}}{4} + \frac{μ^{4}}{81}} - \frac{s^{2}}{2})} - \sqrt[3]{\frac{μ^{4}}{3} (\sqrt{\frac{s^{4}}{4} + \frac{μ^{4}}{81}} + \frac{s^{2}}{2})}}}

Commented by ajfour last updated on 24/Jan/24

u t t e r l y a m a z i n g S i r .

Commented by mr W last updated on 25/Jan/24

t h a n k s s i r! t h e q u e s t i o n 2 i s m u c h

h a r d e r . h a v e y o u t r i e d ?

Commented by ajfour last updated on 25/Jan/24

Thats why i havnt tried, but i m starting to think, few hours...

Commented by mr W last updated on 27/Jan/24

s o l u t i o n t o q u e s t i o n 2

s e e Q 203726

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