Question

A particle at the end of a spring executes simple harmonic motion with a period t₁, while the corresponding period for another spring is t₂. If the period of oscillation with the two springs in series is t, then

T = t₁ + t₂

$straight T squared space equals space straight t subscript 1 superscript 2 space plus space straight t subscript 2 superscript 2$
$space straight T to the power of negative 1 end exponent space equals straight t subscript 1 superscript negative 1 end superscript space plus straight t subscript 2 superscript negative 1 end superscript$
$space straight T to the power of negative 1 end exponent space equals straight t subscript 1 superscript negative 2 end superscript space plus straight t subscript 2 superscript negative 2 end superscript$

Solution

straight T squared space equals space straight t subscript 1 superscript 2 space plus space straight t subscript 2 superscript 2

Time period of spring
$straight T space equals space 2 straight pi square root of straight M over straight k end root$
k, being the force constant of spring. For first spring and for second spring
we have,
$straight T space equals space 2 straight pi square root of straight m over straight k subscript 1 end root space space space....... space left parenthesis straight i right parenthesis straight T space equals space 2 straight pi square root of straight m over straight k subscript 2 end root space......... space left parenthesis ii right parenthesis$
The effective force constant in their series combination is
$straight k space equals fraction numerator straight k subscript 1 straight k subscript 2 over denominator straight k subscript 1 plus straight k subscript 2 end fraction therefore space Time space period space of space combination straight T space equals space 2 straight pi space square root of open square brackets fraction numerator straight m left parenthesis straight k subscript 1 plus straight k subscript 2 right parenthesis over denominator straight k subscript 1 straight k subscript 2 end fraction close square brackets end root straight T squared space equals space fraction numerator 4 straight pi squared straight m space left parenthesis straight k subscript 1 plus straight k subscript 2 right parenthesis over denominator straight k subscript 1 straight k subscript 2 end fraction space...... space left parenthesis iii right parenthesis From space equations space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space obtain straight t subscript 1 superscript 2 space plus space straight t subscript 2 superscript 2 space equals space 4 straight pi squared space open parentheses straight m over straight k subscript 1 space plus straight m over straight k subscript 2 close parentheses rightwards double arrow straight t subscript 1 superscript 2 space plus space straight t subscript 2 superscript 2 space equals space 4 straight pi squared straight m space open parentheses 1 over straight k subscript 1 plus 1 over straight k subscript 2 close parentheses straight t subscript 1 superscript 2 space plus straight t subscript 2 superscript 2 space equals space fraction numerator 4 straight pi squared straight m space left parenthesis straight k subscript 1 plus straight k subscript 2 right parenthesis over denominator straight k subscript 1 straight k subscript 2 end fraction straight t subscript 1 superscript 2 space plus straight t subscript 2 superscript 2 space equals space straight T squared$

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