A mass $m$ is suspended separately by two different springs of spring constant $K_1$ and $K_2$ gives the time-period ${t_1}$ and ${t_2}$ respectively. If same mass $m$ is connected by both springs as shown in figure then time-period $t$ is given by the relation
AIPMT 2002, Medium
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(d) ${t_1} = 2\pi \sqrt {\frac{m}{{{K_1}}}} $ and ${t_2} = 2\pi \sqrt {\frac{m}{{{K_2}}}} $
Equivalent spring constant for shown combination is
$K_1 + K_2$. So time period $t$ is given by $t = 2\pi \sqrt {\frac{m}{{{K_1} + {K_2}}}} $
By solving these equations we get ${t^{ - 2}} = t_1^{ - 2} + t_2^{ - 2}$
Equivalent spring constant for shown combination is
$K_1 + K_2$. So time period $t$ is given by $t = 2\pi \sqrt {\frac{m}{{{K_1} + {K_2}}}} $
By solving these equations we get ${t^{ - 2}} = t_1^{ - 2} + t_2^{ - 2}$
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