Pj Problems - Overview
The Number Line
7 Spaces Of Interest - Overview
Triadic Unit Mesh
States Of Matter
COHN - Natures Engineering Of The Human Body
The Human-Body Systems
Faith, Love, Charity
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:
utt = α2uxx ---------------(1)
where u(x,t) is the displacement of a point on the vibrating substance from its equilibrium position.
utt is the second partial derivative of u(x,t) with respect ot t
uxx (concavity) is the second partial derivative of u(x,t) with respect to x
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).
S7P4A44 (Motion - Oscillatory).
Pj Problem of Interest is of type motion (oscillatory).
(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∂2u(x,t)/∂t2. 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 which has a net transverse force on the small subregion of the string as follows:
Tension component = Tsinθ2 - Tsinθ1
= ≈ 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 restoring force is negative and downward if the displacement is positive and above the x-axis.
Now, applying Newton's equation of motion to the small subregion of the string, yields the following equation:
Δxρ∂2u(x,t)/∂t2 = T[∂u(x + Δx,t)/∂x - ∂u(x,t)/∂x] + ΔxF(x,t) -Δxβ∂u(x,t)/∂t - Δxγu(x,t)----(2)
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:
∂2u(x,t)/∂t2 = α2∂2u(x,t)/∂x2 [-β∂u(x,t)/∂t - γu(x,t) + F(x,t)]/ρ----(3).
Equation (3) is known as the telephone equation and is sometimes written without the density ρ (it is implied even if it is not written).
The simple basic wave equation is :
∂2u(x,t)/∂t2 = α2∂2u(x,t)/∂x2----(4)
Equation (4) intuitively infer that the acceleration of each point of the string is due to the tension in the string. The larger the concativity (uxx), the stronger the force (where α2 = T/ρ is the proportionality constant).
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 = x2 About The X Axis
Single Variable Functions
Absolute Value Functions
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
Derivation Of Heat Equation For A One-Dimensional Heat Flow
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.
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