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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=2πml4F0
Let 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 sinθtanθxl 
Frest=2F0xl
Similarly, Restoring force produced by remaining two springs BP and DP is given by F′′rest=2F0xl
Net Restoring Force F=Frest+F′′rest=4F0xl
Comparing the above equation with  F=Keff x, we get Keff=4F0l
Time period of oscillation T=2πmKeff=2πml4F0
Thus, option (c) is the correct answer.

Physics

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