Three identical ideal springs, each of force constant k are joined to three identical balls (each$$ of mass $$m) as shown in the figure. O is centroid of the triangle. Initially, each of the springs is in its natural length. Now all the three balls are simultaneously given small displacements of equal magnitude along the directions shown in the figure. The oscillation frequency of balls is $$12$$πβ$$k2m$$ . The value of β is $$____________$$.

Question

Three identical ideal springs, each of force constant k are joined to three identical balls (each$$ of mass $$m) as shown in the figure. O is centroid of the triangle. Initially, each  of the springs is in its natural length. Now all the three balls are simultaneously given small displacements of equal magnitude along the directions shown in the figure. The oscillation frequency of balls is $$12$$πβ$$k2m$$ . The value of β is $$____________$$.

Infinity Learn · Accepted Answer

In the figure, $$0$$ $$=$$ $$30o$$.When displacement is each ball x, the extension in each spring is x′$$=xcos$$⁡θ$$=x32Each$$ spring is connected to two balls.So, net extension of each spring $$=2x$$′$$=3xForce$$ along the direction of displacement on each ball $$isF=k3xcos$$⁡$$30$$∘$$+k3xcos$$⁡$$30$$∘$$=3kxF=$$−$$3kx$$ [negative because restoring in nature]$$a=Fm=$$−$$3kmx$$⇒ω$$=3kmf=22$$$$π$$$$3km=12$$$$π$$$$6k2m$$⇒β$$=6$$

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