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Question

# A particle of mass m is attached with four identical springs, each of length l. Initially, each spring has tension F0. Neglecting gravity, calculate the time period for small oscillations of the particle along a line perpendicular to the plane of the figure.

A
T=2πmlF0
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B
T=2πml2F0
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C
T=2πml4F0
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D
T=2π4mlF0
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Solution

## The correct option is C T=2π√ml4F0Let us displace the particle slightly by distance x along a line perpendicular to the plane of figure. Since the springs are identical, increase in tension in each spring is the same. Let the increase in tension be dF0. The figure shows the forces exerted by spring AP and CP: Restoring force produced by these two springs is given by 2(F0+dF0)sinθ in the direction opposite to displacement x. For small oscillations, we can neglect dF0. Also, we know θ is small, so sinθ≈tanθ≈xl ∴F′rest=2F0xl Similarly, Restoring force produced by remaining two springs BP and DP is given by F′′rest=2F0xl Net Restoring Force F=F′rest+F′′rest=4F0xl Comparing the above equation with F=−Keff x, we get Keff=4F0l Time period of oscillation T=2π√mKeff=2π√ml4F0 Thus, option (c) is the correct answer.

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