Q.
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 12πβk2m . The value of β is ____________.
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answer is 6.
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Detailed Solution
In the figure, 0 = 30o.When displacement is each ball x, the extension in each spring is x′=xcosθ=x32Each spring is connected to two balls.So, net extension of each spring =2x′=3xForce along the direction of displacement on each ball isF=k3xcos30∘+k3xcos30∘=3kxF=−3kx [negative because restoring in nature]a=Fm=−3kmx⇒ω=3kmf=22π3km=12π6k2m⇒β=6
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