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Question

In the arrangement shown in figure, pulleys are light and springs are ideal k1, k2, k3 and k4 are force constants of the springs. Calculate period of small vertical oscillations of block of mass m.
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Solution

When the mass m is displaced from its mean position by a distance x, let F be the restoring (extra tension) force produced in the string. By this extra tension further elongation in the springs are 2Fk1, 2Fk2, 2Fk3 and 2Fk4 respectively.
Then,
x=2(2Fk1)+2(2Fk2)+2(2Fk3)+2(2Fk4)
or F(2k1+2k2+2k3+2k4)=x
Here netative sign spows the restoring nature of force.
a=xm(4k1+4k2+4k3+4k4)
T=2πxa
=4πm(1k1+1k2+1k3+1k4)

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