# Four springs Oscillation

**1. The problem statement, all variables and given/known data**

Figure shows a particle of mass m attached with four identical springs each of length L_{0}. Initial tension in each spring is F_{0}.Neglect gravity calculate the period of small oscillations of the particle along a line perpendicular to the plane of figure. (Take F_{0} = (0.01 π^{2})N, m = 100 gm, L_{0} = 10cm)

**2. Relevant equations**

**3. The attempt at a solution**

In the equilibrium state length of the spring L = L_{0} + z ,where L_{0} is the original length and Z is the initial extension in the spring when in equilibrium.

Initial tension in each spring when in equilibrium = kz = F_{0}

Suppose the mass is displaced by x amount upwards .In the attachment L’ represents the new length.

L’=√(L^{2}+x^{2)}

Force by a spring = k(L’-L_{0}) = = k(L’-L+z) = k(L’-L)+kz

This force will have a a horizontal component and a a vertical component .The horizontal component will be cancelled by a similar component from the opposite string .

The net force from all four springs will be F = 4k(L’-L_{0} )sinθ = 4k(L’-L)sinθ + 4kzsinθ

sinθ = x/L’

So net force =4kx(L’-L)/L’ + 4kxz/L

=4kx(1-L/L’) + 4kxz/L

I somehow feel I am not approaching the problem correctly .

I would be grateful if somebody could help me with the problem.

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