Physics-
General
Easy

Question

Infinite springs with force constants k comma 2 k comma 4 k a n d blank 8 k horizontal ellipsis.. respectively are connected in series. The effective force constant of the spring will be

  1. 2 k    
  2. k    
  3. k divided by 2    
  4. 2048    

The correct answer is: k divided by 2


    Effective force constant is equal to the reciprocal of the sum of individual force constant, hence
    fraction numerator 1 over denominator k subscript e end subscript end fraction equals fraction numerator 1 over denominator k subscript 1 end subscript end fraction plus fraction numerator 1 over denominator k subscript 2 end subscript end fraction plus fraction numerator 1 over denominator k subscript 3 end subscript end fraction plus horizontal ellipsis
    Given, k subscript 1 end subscript equals k comma blank k subscript 2 end subscript equals 2 k comma blank k subscript 3 end subscript equals 3 k comma horizontal ellipsis..
    therefore fraction numerator 1 over denominator k subscript e end subscript end fraction equals fraction numerator 1 over denominator k end fraction plus fraction numerator 1 over denominator 2 k end fraction plus fraction numerator 1 over denominator 4 k end fraction plus fraction numerator 1 over denominator 8 k end fraction plus horizontal ellipsis
    The given series a geometric progression series, hence sum is
    S subscript infinity end subscript equals fraction numerator a over denominator 1 minus r end fraction
    where a is first term of series and r the common difference.
    rightwards double arrow fraction numerator 1 over denominator k subscript e end subscript end fraction equals fraction numerator 1 over denominator k end fraction cross times fraction numerator 1 over denominator open parentheses 1 minus fraction numerator 1 over denominator 2 end fraction close parentheses end fraction equals fraction numerator 2 over denominator k end fraction rightwards double arrow k subscript e end subscript equals fraction numerator k over denominator 2 end fraction

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