A helical spring of stiffness \(k\) is cut into two halves and a mass \(m\) is connected to the two halves as shown in Fig. 2.81(a). The natural time period of this system is found to be \(0.5 \mathrm{~s}\). If an identical spring is cut so that one part is one-fourth and the other part three-fourths of the original length, and the mass \(m\) is connected to the two parts as shown in Fig. 2.81(b), what would be the natural period of the system?

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+ 00000 m 00000 (a) FIGURE 2.81 Mass connected to springs in two ways. m 00000000 (b)