## Simple Harmonic Motion: A Comprehensive Guide

## 1) Definition of Linear Simple Harmonic Motion

Linear Simple Harmonic Motion (SHM) is a periodic motion where the restoring force (or acceleration) is always directed towards the mean position and its magnitude is directly proportional to the displacement from the mean position.

## 2) Differential Equation of Linear SHM

The differential equation of SHM is:

^{2}x +ω

^{2.}x = 0 …….(1)

^{2}

### (a) Expression for Acceleration in SHM

^{2}x =

^{-ω2.x…….[From(1)]}

^{2}

But a =d^{2}x is the acceleration of the particle

## Fluid Mechanics: Exploring Bernoulli’s Theorem, Boundary Layer Separation, and Fluid Properties

## Fluid Mechanics

### Bernoulli’s Theorem for Streamline Flow

**where Cb = Coefficient of venturimeter and its value is less than 1**

Bernoulli’s theorem states that in a steady flow of an incompressible and non-viscous fluid along a streamline, the sum of pressure energy, kinetic energy, and potential energy per unit mass remains constant.

#### Proof:

#### Pressure Energy:

The pressure energy per unit mass is given by (P/ρ), where (P) is the pressure and (ρ) is the density of the fluid. This represents the work done

Read More## Area-Velocity Relationship in Compressible Fluid Flow: Derivation and Explanation

## Area-Velocity Relationship for Compressible Fluid Flow

To derive the area-velocity relationship for a compressible fluid flow, we use the principles of conservation of mass (continuity equation) and the concept of isentropic flow (assuming the flow is adiabatic and reversible). This relationship helps us understand how changes in the cross-sectional area of a flow affect the velocity in a compressible fluid flow.

### 1. Continuity Equation

For a compressible fluid, the continuity equation (conservation

Read More## Introduction to Nanotechnology: Properties, Applications, and Quantum Effects

## Nanotechnology

Nano means 10^{-9}. A nanometer is 1 billionth or 10^{-9} of a meter. Nanomaterials are materials with structured components smaller than 100nm.

1nm = 10^{-9}m

### Properties of Nanoparticles

Many properties of solids depend on their size. Nanoscale materials exhibit significantly different properties compared to their larger counterparts due to two primary factors:

**Increased surface area to volume ratio:**This enhances reactivity, catalytic activity, and influences electrical, thermal, and mechanical

## Electromagnetism

## Q # 1. The potential is constant throughout a given region of space. Is the electric field zero or non zero in this region? Explain.

### Ans.

The electric field intensity is described by the relation:

According to the relation, the electric field is negative gradient of electric potential. If the electric

potential is constant throughout given region of space, then change in electric potential , hence .

## Q # 2. Suppose that you follow an electric field line due to a positive point charge. Do electric field

Read More## Scientific Discoveries: A Journey of Serendipity and Innovation

## Scientific Discoveries: A Journey of Serendipity and Innovation

### Electromagnetism

In 1819, Danish physicist H. C. Ørsted made a groundbreaking discovery while conducting a classroom demonstration. As he passed an electric current through a copper wire connected to a voltaic cell, he noticed that a nearby magnetic needle deflected. This observation led him to the discovery of electromagnetism, a fundamental principle in the field of electricity.

### Vulcanization

American inventor Charles Goodyear embarked