Expressions Of Pj Problems

Pj Problems - Overview

Celestial Stars

The Number Line

Geometries

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation

The Atom

Survival

Energy

Light

Heat

Sound

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Language

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States Of Matter

Buoyancy

Nuclear Reactions

Molecular Shapes

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Human-cells

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COHN - Natures Engineering Of The Human Body

The Human-Body Systems

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Photosynthesis

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Differential Calculus

Antiderivative

Integral Calculus

Economies

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Painting

Derivation Of The One Dimensional Wave Equation

There are three basic types of Linear Partial Differential Equations (PDEs). *Parobolic PDEs, Hyperbolic PDEs and Elliptic PDES*. The one dimensional wave equation is a hyperbolic PDE and is of the form:
**u _{tt} = α^{2}u_{xx}** ---------------(1)

where u(x,t) is the displacement of a point on the vibrating substance from its equilibrium position.

u

u

α is the proportinality constant.

Show that the transverse vibrations of a string of length L (figure 114.8a) fastened at each end can be described mathemathecally by equation (1).

**The strings**:
S_{7}P_{4}A_{44} (Motion - Oscillatory).
**The math**:

Pj Problem of Interest is of type *motion* (oscillatory).

Assumptions:

(1) The string is fastened at each end and stretched tightly.

(2) The string is made of homogeneous material.

(3) The string is unaffected by gravity.

(4) The vibrations take place in a plane

Consider a small subregion [x, x + Δ] of the vibrating string. Figure 114.8b is a magnification of this small subregion.

According to Newton's equation of motion:

Change in momentum of the small subregion of the string is equal to the applied forces
**The change in momentum = m∂ ^{2}u(x,t)/∂t^{2}**. Where m is mass.

Important forces acting on the string:

(1) Net force due to the tension in the string = T ( figure 114.8b)

T has a

Tension component = Tsinθ

= ≈ T[∂u(x + Δx,t)/∂x - ∂u(x,t)/∂x]

(2) External force F(x,t) = -mg (gravity).

(3) Frictional force (- β∂u(x,t)/∂t))

This is a resistance force from the medium in which the string is vibrating at a velocity of ∂u(x,t)/∂t).

(4) Restoring force (γu(x,t)).

This is the force acting opposite to the displacement of the string. The

Now, applying

Where ρ is the density of the string.

Dividing equation (2) by ρ and Δx and letting Δx ----> 0 (tends to 0), results in the following equation:

Equation (3) is known as the

The simple basic wave equation is :

Equation (4) intuitively infer that the acceleration of each point of the string is due to the tension in the string. The larger the

Math

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

Derivation Of The Area Of A Circle, A Sector Of A Circle And A Circular Ring

Derivation Of The Area Of A Trapezoid, A Rectangle And A Triangle

Derivation Of The Area Of An Ellipse

Derivation Of Volume Of A Cylinder

Derivation Of Volume Of A Sphere

Derivation Of Volume Of A Cone

Derivation Of Volume Of A Torus

Derivation Of Volume Of A Paraboloid

Volume Obtained By Revolving The Curve y = x^{2} About The X Axis

Single Variable Functions

Absolute Value Functions

Conics

Real Numbers

Vector Spaces

Equation Of The Ascent Path Of An Airplane

Calculating Capacity Of A Video Adapter Board Memory

Probability Density Functions

Boolean Algebra - Logic Functions

Ordinary Differential Equations (ODEs)

Infinite Sequences And Series

Introduction To Group Theory

Advanced Calculus - Partial Derivatives

Advanced Calculus - General Charateristics Of Partial Differential Equations

Advanced Calculus - Jacobians

Advanced Calculus - Solving PDEs By The Method Of Separation Of Variables

Advanced Calculus - Fourier Series

Advanced Calculus - Multiple Integrals

Production Schedule That Maximizes Profit Given Constraint Equation

Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation

Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions

Fourier Series

Derivation Of Heat Equation For A One-Dimensional Heat Flow

Homogenizing-Non-Homogeneous-Time-Varying-IBVP-Boundary-Condition

The Universe is composed of *matter* and *radiant energy*. *Matter* is any kind of *mass-energy* that moves with velocities less than the velocity of light. *Radiant energy* is any kind of *mass-energy* that moves with the velocity of light.

Periodic Table

Composition And Structure Of Matter

How Matter Gets Composed

How Matter Gets Composed (2)

Molecular Structure Of Matter

Molecular Shapes: Bond Length, Bond Angle

Molecular Shapes: Valence Shell Electron Pair Repulsion

Molecular Shapes: Orbital Hybridization

Molecular Shapes: Sigma Bonds Pi Bonds

Molecular Shapes: Non ABn Molecules

Molecular Orbital Theory

More Pj Problem Strings