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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 $fraction numerator f subscript s end subscript over denominator f subscript p end subscript end fraction$ ?

1:2
2:1
1:3
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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