In the arrangement shown in figure, pulleys are light and springs are ideal $k_{1}$ , $k_{2}$ , $k_{3}$ and $k_{4}$ are force constants of the springs. Calculate period of small vertical oscillations of block of mass m.

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Solution

When the mass m is displaced from its mean position by a distance x, let F be the restoring (extra tension) force produced in the string. By this extra tension further elongation in the springs are $\frac{2 F}{k_{1}}$ , $\frac{2 F}{k_{2}}$ , $\frac{2 F}{k_{3}}$ and $\frac{2 F}{k_{4}}$ respectively.
Then,
$x = 2 (\frac{2 F}{k_{1}}) + 2 (\frac{2 F}{k_{2}}) + 2 (\frac{2 F}{k_{3}}) + 2 (\frac{2 F}{k_{4}})$
or $F (\frac{2}{k_{1}} + \frac{2}{k_{2}} + \frac{2}{k_{3}} + \frac{2}{k_{4}}) = - x$
Here netative sign spows the restoring nature of force.
$a = - \frac{x}{m (\frac{4}{k_{1}} + \frac{4}{k_{2}} + \frac{4}{k_{3}} + \frac{4}{k_{4}})}$
$T = 2 π \sqrt{∣ ∣ \frac{x}{a} ∣ ∣}$
$= 4 π \sqrt{m (\frac{1}{k_{1}} + \frac{1}{k_{2}} + \frac{1}{k_{3}} + \frac{1}{k_{4}})}$

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In the arrangement shown in figure, pulleys are light and springs are ideal k1, k2, k3 and k4 are force constants of the springs. Calculate period of small vertical oscillations of block of mass m.

In the arrangement shown in figure, pulleys are light and springs are ideal $k_{1}$ , $k_{2}$ , $k_{3}$ and $k_{4}$ are force constants of the springs. Calculate period of small vertical oscillations of block of mass m.