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): 
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
Stereoisomers occur when the ligands have the same bonds, but the bonds are in different orientations relative to one another. Structural isomerism occurs when the bonds are different. Isomers are distinct compounds that can have different physical properties such as color, crystal structure, and melting point.
In octahedral complexes—with four of one ligand and two of another—and square planar complexes—with two of one ligand and two of another—there are two different arrangements of the same atoms with the same bonds. These different arrangements are called cis and trans. In cis molecules, the two ligands are on the same side of the complex. In trans molecules, the similar ligands are on the opposite sides of the molecules, meaning the bond angle is 180 degrees.
Optical isomerism occurs when a molecule is not superimposable with its mirror image.
When three identical ligands occupy one face of an octahedron, the isomer is said to be facial or fac. If these three ligands and the metal ion are in one plane, the isomer is said to be meridional or mer.