The center of a disk of radius $ r $ and mass $ m $ is attached to a spring of spring constant $ k $, inside a ring of radius $ R>r $ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $ T = \frac{2\pi}{\omega} $. The correct expression for $ \omega $ is ( $ g $ is the acceleration due to gravity):
Match List-I with List-II
\[\begin{array}{|l|l|} \hline \text{List-I (Soil component)} & \text{List-II (Definition)} \\ \hline (A)~\text{Azonal soils} & (I)~\text{An individual natural aggregate of soil particles.} \\ (B)~\text{Regoliths} & (II)~\text{Organisms living in the soil or ground} \\ (C)~\text{Ped} & (III)~\text{Soils have uniformity from the top-surface to the base, and do not have well-developed soil horizons.} \\ (D)~\text{Edaphons} & (IV)~\text{Zone of loose and unconsolidated weathered rock materials.} \\ \hline \end{array}\]
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Match List-I with List-II
\[\begin{array}{|l|l|} \hline \text{List I Content of humus} & \text{List II Percentage of contents} \\ \hline \text{(A) Carbon} & \text{(I) 35-40\%} \\ \hline \text{(B) Oxygen} & \text{(II) ~5\%} \\ \hline \text{(C) Hydrogen} & \text{(III) 55-60\%} \\ \hline \text{(D) Nitrogen} & \text{(IV) 15\%} \\ \hline \end{array}\]
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