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Three pulley system

The pulley, strings and springs are light.I just wanted to confirm my answer (although I know I am wrong) that

\[\Large{\omega^2=\frac{4}{M \left(\frac{1}{k_{1}}+\frac{2}{k_{2}}+\frac{4}{k_{3}}\right)}}\]

Note by Tanishq Varshney
8 months, 4 weeks ago

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i'm getting \(\large\frac{4}{M(\frac{1}{k_{1}}+\frac{1}{4k_{2}}+\frac{1}{16k_3})}\) Keshav Tiwari · 8 months, 4 weeks ago

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@Keshav Tiwari Yeah. Pulkit Gupta · 8 months, 3 weeks ago

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Comment deleted 8 months ago

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@Aditya Kumar Sorry I don't have its answer can u post your method/solution please! Tanishq Varshney · 8 months, 3 weeks ago

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@Keshav Tiwari @Aditya Kumar @Abhineet Nayyar plz post your method. Tanishq Varshney · 8 months, 3 weeks ago

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@Tanishq Varshney Take the tension with the string attached to the mass as \(8T\). Now, the following strings will have half the tension as the previous one, as you already know. Consider the mass moves by a distance \(x\). The spring \({ k }_{ 1 }\) gets stretched by \({ x }_{ 1 }\), \({ k }_{ 2 }\) with \({ x }_{ 2 }\), and \({ k }_{ 3 }\) with \({ x }_{ 3 }\). You can clearly see, that \({ k }_{ 1 }\times { x }_{ 1 }=4T\), \({ k }_{ 2 }\times { x }_{ 2 }=2T\) and \({ k }_{ 3 }\times { x }_{ 3 }=T\) Also by virtual work done: \(8x=4{ x }_{ 1 }+2{ x }_{ 2 }+{ x }_{ 3 }\)

Put \({ x }_{ 1 }=4a/{ k }_{ 1 }\), \({ x }_{ 2 }=2a/{ k }_{ 2 }\), and \({ x }_{ 3 }=a/{ k }_{ 3 }\). Also, put \(x=a/k\) where k is the net inertia factor and \(x, k\) will also follow the same relation as the other pairs.

Solving this: \({ \omega }^{ 2 }=\frac { 8 }{ M(\frac { 16 }{ { k }_{ 1 } } +\frac { 4 }{ { k }_{ 2 } } +\frac { 1 }{ { k }_{ 3 } } ) } \) Abhineet Nayyar · 8 months, 3 weeks ago

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I'm getting \(\frac { 8 }{ M(\frac { 16 }{ { k }_{ 1 } } +\frac { 4 }{ { k }_{ 2 } } +\frac { 1 }{ { k }_{ 3 } } ) } \) Abhineet Nayyar · 8 months, 3 weeks ago

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