Two identical springs (spring constant = k ) are connected to two light rigid rods in parallel combination and then a third spring (identical to the other two) is connected to this combination in seri

Question

Two identical springs (spring constant = $k$ ) are connected to two light rigid rods in parallel combination and then a third spring (identical to the other two) is connected to this combination in series. The combined system is fixed to the ceiling and a block of mass $m$ is attached as shown in the figure. If the mass is slightly displaced and then let go, then the system shall oscillate with a time period of

TestHub · Accepted Answer

Consider the system as shown in figure below.

Here, $k_{2}$ and $k_{3}$ are in parallel. $k_{1}$ is in series with parallel combination of $k_{2}$ and $k_{3} .$

Effective spring constant of the combination is

$k_{e f f} = \frac{(k_{2} + k_{3}) k_{1}}{k_{1} + k_{2} + k_{3}}$

$= \frac{2 k \times k}{3 k} = \frac{2}{3} k$

$∴$ Time period of oscillation

$T = 2 π \sqrt{\frac{m}{k_{e f f}}}$

$= 2 π \sqrt{\frac{m}{\frac{2}{3} k}}$

$= 2 π \sqrt{\frac{3 m}{2 k}}$

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