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Algebra of Limits

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Question

Find the constraint relation between $a_{1} + a_{2}$ and $a_{3} .$

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Solution

We will try to find the length of rope, that's a constant quantity.
As we can see from the figure $x_{1} + x_{4} = C$ .......(1) C is constant
Also $(x_{2} - x_{1}) + (x_{3} - x_{4}) = C_{1}$ ...(2)
On double differentiating the above equation
$a_{1} + a_{4} = 0$
$a_{2} + a_{3} - 2 a_{4} = 0$
eliminating the value of $a_{4}$ i.e. $a_{4} = - a_{1}$
We get $a_{3} + a_{2} + 2 a_{1} = 0$

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