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A particle at the end of a spring executes SHM with a period t_(1), while corresponding period for another spring is t_(2). If the period of oscillation when the two springs are connected in series is T, then prove that, t_(1)^(2)+t_(2)^(2)=T^(2).

Answer» <html><body><p></p>Solution :<a href="https://interviewquestions.tuteehub.com/tag/let-11597" style="font-weight:bold;" target="_blank" title="Click to know more about LET">LET</a> the mass of the particle be m and <a href="https://interviewquestions.tuteehub.com/tag/spring-11487" style="font-weight:bold;" target="_blank" title="Click to know more about SPRING">SPRING</a> constants of the <a href="https://interviewquestions.tuteehub.com/tag/springs-17108" style="font-weight:bold;" target="_blank" title="Click to know more about SPRINGS">SPRINGS</a> be `k_(1)andk_(2)`. <br/> In first case, <br/> `t_(1)=2pisqrt(m/k_(1))or,t_(1)^(2)=4pi^(2)(m/k_(1))""...(1)` <br/> In second case, <br/> `t_(2)=2pisqrt(m/k_(2))or,t_(2)^(2)=4pi^(2)(m/k_(2))""...(2)` <br/> In series the equivalent spring <a href="https://interviewquestions.tuteehub.com/tag/constant-930172" style="font-weight:bold;" target="_blank" title="Click to know more about CONSTANT">CONSTANT</a> is k, <br/> Then, `1/k=1/k_(1)+1/k_(2)or,k=(k_(1)k_(2))/(k_(1)+k_(2))` <br/> `therefore` Time <a href="https://interviewquestions.tuteehub.com/tag/period-1151023" style="font-weight:bold;" target="_blank" title="Click to know more about PERIOD">PERIOD</a> of the combination, <br/> `T=2pisqrt(m/k)=2pisqrt((m(k_(1)+k_(2)))/(k_(1)k_(2)))` <br/> or, `T^(2)=(4pi^(2)m(k_(1)+k_(2)))/(k_(1)k_(2))` <br/> adding (1) and (2) we get, <br/> `t_(1)^(2)+t_(2)^(2)=4pi^(2)(m/k_(1)+m/k_(2))=4pi^(2)m(1/k_(1)+1/k_(2))` <br/> `=4pi^(2)m((k_(1)+k_(2))/(k_(1)k_(2)))` <br/> `therefore" "t_(1)^(2)+t_(2)^(2)=T^(2)` (Proved).</body></html>


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