The springs shown in the figure 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.

$\frac{F (k_{2} + k_{3})}{{k_{1}}^{2} {k_{2}}^{2} + {k_{3}}^{3}}, \frac{1}{2 π} \sqrt{\frac{{k_{1}}^{2} {k_{2}}^{2} + {k_{3}}^{3}}{M (k_{2} + k_{3})}}$

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

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

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Solution

The correct option is C
$\frac{F (k_{2} + k_{3})}{k_{1} k_{2} + k_{2} k_{3} + k_{1} + k_{3}}, \frac{1}{2 π} \sqrt{\frac{k_{1} k_{2} + k_{2} k_{3} + k_{1} + k_{3}}{M (k_{2} + k_{3})}}$

Spring $k_{2} a n d k_{3}$ are in series their combination, say $k_{4}$ would be

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

now $k_{4} a n d k_{1}$ are in parallel

$k_{5} = k_{1} + k_{4} = k_{1} + \frac{k_{2} k_{3}}{k_{2} + k_{3}}$
$= \frac{k_{1} k_{2} + k_{2} k_{3} + k_{1} k_{3}}{k_{2} + k_{3}}$
$f = \frac{1}{2 π} \sqrt{\frac{k}{m}} = \frac{1}{2 π} \sqrt{\frac{k_{1} k_{2} + k_{2} k_{3} + k_{1} k_{3}}{M (k_{2} + k_{3})}}$
now mean position, $F = k_{5} x$
$x = \frac{F (k_{2} + k_{3})}{k_{1} k_{2} + k_{2} k_{3} + k_{1} k_{3}}$
that would be the amplitude of new motion

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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.