The springs shown in the figure (12-E7) are all unstretched in the beginning when a man starts pulling the block. The man exerts a constant force F on the block. Find the amplitude and the frequency of the motion of the block.

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Solution

$k_{2}$ and $k_{3}$ are in series.

Let equivalent spring const. be $k_{4} .$

$∴ \frac{1}{k_{4}} = \frac{1}{k_{2}} + \frac{1}{k_{3}} = \frac{k 2 + k_{3}}{k_{2} k_{3}}$

$k_{4} = \frac{k_{2} k_{3}}{k_{2} + k_{3}}$

Now, $k_{4}$ and $k_{1}$ are in parallel. So equivalent spring constant $k = k_{1} + k_{4}$

$= \frac{k_{2} k_{3}}{k_{2} + k_{3}} + k_{1}$

$= \frac{k_{2} k_{3} + k_{1} k_{2} + k_{1} k_{3}}{k_{2} + k_{3}}$

$∴ T = 2 π \sqrt{\frac{M}{k}}$

$= 2 π \sqrt{\frac{M (k_{2} + k_{3})}{k_{2} k_{3} + k_{1} k_{2} + k_{1} k_{3}}}$

(b) Frequency = $\frac{1}{T}$

$= \frac{1}{2 π} \sqrt{\frac{k_{2} + k_{3} + k 1 k 2 + k 1 k + 3}{M (k_{2} + k_{3})}}$

(c) Amplitude = x

$= \frac{F}{k} = \frac{F (k_{2} + k_{3})}{k_{1} k_{2} + k_{2} k_{3} + k_{1} k_{3}}$

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