
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 ________.
Diodes are used in many applications. By removing the signal component, several of them convert AC current to DC current also are known as rectifiers when used in this position. Diodes act as electric switches and can block voltage spikes, making them ideal for surgical protectors. They are hired to do digital logic. Some things are made from diodes such as power supply and voltage doubles. The sensors, as well as light on lighting devices and lasers, depend on LEDs. For electronic tuning, varactors are used, and for compressing transients in AC lines varistors are used. Zener diodes are used as voltage regulators where varactors are used for electronic tuning, and varistors are used for compressing transients in AC lines.
On the base of diodes, transistors and op-amps are built. The most common diode mode is the p-n junction. One (n) object with electrons as the charging conductor closes the second object (p) with holes (well-charged particles acted as depleted areas of electrons) as the charge carriers on this type of diode. Where they connect, the narrowing space forms, where electrons are dispersed to fill holes in the p-side. This effectively stops the flow of electrons. When a positive voltage is given to the p-side of this junction, electrons may easily move from it and the current flows into the diode to fill the holes. The shrinkage area expands as the negative bias interacts (i.e., negative voltage is given to the p-side), making the electrons harder to move across.