Find the time period of the oscillation of mass m in figures 12-E4 a,b,c. What is the equivalent spring constant of the pair of springs in each case ?

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Solution

(a) Equivalent spring constant,

$k = k_{1} + k_{2}$ (parallel)

$T = 2 π \sqrt{\frac{m}{k}} = 2 π \sqrt{\frac{m}{k_{1} + k_{2}}}$

(b) Let us, displace the block m towards left through displacement 'x'.

Resultant force,

$F = F_{1} + F_{2} = (k_{1} + k_{2}) x$

Acceleration,

$(\frac{F}{m}) = \frac{(k_{1} + k_{2})}{m} x$

Time period,

$T = 2 π \sqrt{\frac{displacement}{Acceleration}}$

$= 2 π \sqrt{\frac{x}{x \frac{(k_{1} + k_{2})}{m}}}$

$= 2 π \sqrt{\frac{m}{k_{1} + k_{2}}}$

The equivalent spring constant

$k = k_{1} + k_{2}$

(c) In series connection equivalent spring constant be k.

So, $\frac{1}{k} = \frac{1}{k_{1}} + \frac{1}{k_{2}} = \frac{k_{2} + k + 1}{k_{1} k_{2}}$

$\Rightarrow K = \frac{k_{1} k_{2}}{k_{1} + k_{2}}$

$T = 2 π \sqrt{\frac{m}{k}}$

$= 2 π \sqrt{\frac{m (k_{1} + k_{2})}{k_{1} k_{2}}}$

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