Three pulley system

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

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

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

@Abhishek Singh @Abhineet Nayyar @Abhishek Sharma @Aditya Kumar

Tanishq Varshney - 5 years, 5 months ago

i'm getting $\large\frac{4}{M(\frac{1}{k_{1}}+\frac{1}{4k_{2}}+\frac{1}{16k_3})}$

Keshav Tiwari - 5 years, 5 months ago

Yeah.

Pulkit Gupta - 5 years, 5 months ago

I'm getting $\frac { 8 }{ M(\frac { 16 }{ { k }_{ 1 } } +\frac { 4 }{ { k }_{ 2 } } +\frac { 1 }{ { k }_{ 3 } } ) }$

A Former Brilliant Member - 5 years, 5 months ago

@Keshav Tiwari @Aditya Kumar @Abhineet Nayyar plz post your method.

Tanishq Varshney - 5 years, 5 months ago

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: ${ \omega }^{ 2 }=\frac { 8 }{ M(\frac { 16 }{ { k }_{ 1 } } +\frac { 4 }{ { k }_{ 2 } } +\frac { 1 }{ { k }_{ 3 } } ) }$

A Former Brilliant Member - 5 years, 5 months ago

Are you sure that you didn't exchange the indexes 1 and 3?

Gabriele Manganelli - 3 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$