Physics-
General
Easy

Question

Two identical springs are connected in series and parallel as shown in the figure. If f subscript s end subscript a n d f subscript p end subscript are frequencies of arrangements, what is fraction numerator f subscript s end subscript over denominator f subscript p end subscript end fraction?

  1. 1:2    
  2. 2:1    
  3. 1:3    
  4. 3:1    

The correct answer is: 1:2


    In first case, springs are connected in parallel, so their equivalent spring constant
    k subscript p end subscript equals k subscript 1 end subscript plus k subscript 2 end subscript
    So, frequency of this spring-block system is
    f subscript p end subscript equals fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator k subscript p end subscript over denominator m end fraction end root
    or f subscript p end subscript equals fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator k subscript 1 end subscript plus k subscript 2 end subscript over denominator m end fraction end root
    but k subscript 1 end subscript equals k subscript 2 end subscript equals k
    therefore f subscript p end subscript equals fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator 2 k over denominator m end fraction end root..(i)
    Now in second case, springs are connected in series, so their equivalent spring constant
    k equals fraction numerator k subscript 1 end subscript k subscript 2 end subscript over denominator k subscript 1 end subscript plus k subscript 2 end subscript end fraction
    Hence, frequency of this arrangement is given by
    f subscript s end subscript equals fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator k subscript 1 end subscript k subscript 2 end subscript over denominator left parenthesis k subscript 1 end subscript plus k subscript 2 end subscript right parenthesis m end fraction end root
    or f subscript s end subscript equals fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator k over denominator 2 m end fraction end root…(ii)
    Dividing Eq. (ii) by Eq. (i), we get
    fraction numerator f subscript s end subscript over denominator f subscript p end subscript end fraction equals fraction numerator fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator k over denominator 2 m end fraction end root over denominator fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator 2 k over denominator m end fraction end root end fraction equals square root of fraction numerator 1 over denominator 4 end fraction end root
    or fraction numerator f subscript s end subscript over denominator f subscript p end subscript end fraction equals fraction numerator 1 over denominator 2 end fraction

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