# JEE Main & Advanced Physics Simple Harmonic Motion Oscillation of Spring Combination

Oscillation of Spring Combination

Category : JEE Main & Advanced

(1) Series combination : If two springs of spring constants ${{K}_{1}}$ and ${{K}_{2}}$ are joined in series as shown then

(i) In series combination equal forces acts on spring but extension in springs are different.

(ii) Spring constants of combination

$\frac{1}{{{k}_{s}}}=\frac{1}{{{k}_{1}}}+\frac{1}{{{k}_{2}}}$$\Rightarrow$${{k}_{s}}=\frac{{{k}_{1}}{{k}_{2}}}{{{k}_{1}}+{{k}_{2}}}$

(iii) If n springs of different force constant are connected in series having force constant ${{k}_{1}},{{k}_{2}},{{k}_{3}}.......$ respectively then

$\frac{1}{{{k}_{S}}}=\frac{1}{{{k}_{1}}}+\frac{1}{{{k}_{2}}}+\frac{1}{{{k}_{3}}}+........$

If all spring have same spring constant then ${{k}_{S}}=\frac{k}{n}$

(iv) Time period of combination $T=2\pi \sqrt{\frac{m}{{{k}_{S}}}}=2\pi \sqrt{\frac{m({{k}_{1}}+{{k}_{2}})}{{{k}_{1}}{{k}_{2}}}}$

(2) Parallel combination : If the springs are connected in parallel as shown

(i) In parallel combination different forces acts on different  springs but extension in springs are same

(ii) Spring constants of combination ${{k}_{P}}={{k}_{1}}+{{k}_{2}}$

(iii) If n springs of different force constant are connected in parallel having force constant ${{k}_{1}},{{k}_{2}},{{k}_{3}}.......$ respectively then ${{k}_{P}}={{k}_{1}}+{{k}_{2}}+{{k}_{3}}+...$

If all spring have same spring constant then ${{k}_{P}}=nk$

(iv) Time period of combination ${{T}_{P}}=2\pi \sqrt{\frac{m}{{{k}_{P}}}}=2\pi \sqrt{\frac{m}{({{k}_{1}}+{{k}_{2}})}}$

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