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Spring Block's Time Period

A block of ma...

Question

A block of mass m is attached to one end of a light inextensible string passing over a smooth light pulley B and under another smooth light pulley A as shown in the figure. The other end of the string is fixed to a ceiling. A and B are held by springs of spring constants $k_{1}$ and $k_{2}$ . Find angular frequency of small oscillations of the system.

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Solution

Let F be the restoring force (extra tension) on block m when displaced by x from its equilibrium position.
$x = 2 x_{1} + 2 x_{2}$
$= 2 [\frac{2 F}{k_{1}} + \frac{2 F}{k_{2}}]$
$= 4 F (\frac{k_{1} + k_{2}}{k_{1} k_{2}})$
or $F = - \frac{k_{1} k_{2}}{4 (k_{1} + k_{2})} x$
$∴$ $a = - \frac{k_{1} k_{2}}{4 m (k_{1} + k_{2})} x$
$ω = \sqrt{\frac{k_{1} k_{2}}{4 m (k_{1} + k_{2})}}$

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