
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 ________.
The study of the chemical phenomena that occur at the interface of two surfaces which can be solid-liquid, solid-gas, solid-vacuum, liquid-gas, etc.
Read Also: Surface Chemistry
Applications of surface chemistry are:
The process of attraction and aggregation of the molecules of a substance on the surface of a solid is known as adsorption. For Example, N2 adsorbs on the surface of activated charcoal
Two types of adsorption are:-
The process through which the refined metals convert themselves into more stable compounds is known as corrosion.
The type of technique used in order to purify the substances to separate Solids from liquids is known as crystallisation.
The process wherein a catalyst is used in order to increase the rate of a chemical reaction is known as catalysis. The catalyst does not undergo any transformation and can be recovered in a chemically unchanged state.
There are two types of catalysis:-