Question

A spring of force constant k is cut into lengths of ratio 1: 2 : 3. They are connected in series and the new force constant is k'. Then they are connected in parallel and force constant is k''. Then k' : k'' is

1:6

1:9

1:11

1:14

Solution

1:11

Sping space constant space proportional to space 1 over length straight k proportional to space 1 over straight l straight i. straight e. space straight k subscript 1 space equals space 6 straight k straight k subscript 2 space equals space 3 straight k straight k subscript 3 space equals space 2 straight k In space series space fraction numerator 1 over denominator straight k apostrophe end fraction space equals space fraction numerator 1 over denominator 6 straight k end fraction space plus space fraction numerator 1 over denominator 3 straight k end fraction space plus space fraction numerator 1 over denominator 2 straight k end fraction fraction numerator 1 over denominator straight k apostrophe end fraction space equals space fraction numerator 6 over denominator 6 straight k end fraction straight k apostrophe space equals space straight k straight k apostrophe apostrophe space equals space 6 straight k space plus space 3 straight k plus space 2 straight k straight k apostrophe apostrophe space equals space 11 straight k fraction numerator straight k apostrophe over denominator straight k apostrophe apostrophe end fraction space equals 1 over 11 1 colon 11

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