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

Music

Language

Stories

Work

States Of Matter

Buoyancy

Nuclear Reactions

Molecular Shapes

Electron Configurations

Chemical Bonds

Energy Conversion

Chemical Reactions

Electromagnetism

Continuity

Growth

Human-cells

Proteins

Nucleic Acids

COHN - Natures Engineering Of The Human Body

The Human-Body Systems

Vision

Walking

Behaviors

Sensors Sensings

Beauty

Faith, Love, Charity

Photosynthesis

Weather

Systems

Algorithms

Tools

Networks

Search

Differential Calculus

Antiderivative

Integral Calculus

Economies

Inflation

Markets

Money Supply

Painting

Resonant Frequency

**(a)** Define:

(i) Waves (ii) Amplitude of a wave (iii) Wavelength (iv) Frequency of a wave
(v) Speed of a wave

(vi) Transverse waves (vii) Longitudinal waves (viii) Stationary waves (ix) Resonant frequency

(x) Resonance (xi) Wave Reflection (xii) Wave refraction (xiii) Wave diffraction

(xiv) Wave interference (xv) Combination of waves.
**(b)** Calculate the resonant frequency of the circuit illustrated in figure 7.7.

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

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

**(ai)** *Waves*: moving *disturbance* that transport *energy* from one place to another. Some waves (*mechanical waves*) need a *medium* in which they propagate, some (*electromagnetic waves*) do not. For example, ocean waves need water as medium. While light waves can propagate with or without a medium.

(ii) *Amplitude of a wave*: maximum vertical distance from rest to crest (figure 7.6)

(iii) *Wavelength*: distance between two consecutive crests or troughs

(iv)*Frequency*: number of complete wave or number of cycles per unit time (usually per sec)

(v) *Speed*: frequency (f) x wavelength (λ)

(vi) *Transverse wave*: motion of *medium* is perpendicular to the direction of wave propagation.

(vii) *Longitudinal wave*: motion of medium is parallel to the direction of wave propagation.

(viii) *Stationary waves* (Standing waves): waves that appear motionless.

(ix) *Resonant frequency* (Natural frequency): frequency at which standing waves occur.

(x) *Resonance*: ability of an object to vibrate by absorbing the energy of its own frequency. Consequently, an object will vibrate if it is in close proximity to another vibrating object having a natural frequency equal to its natural frequency.

(xi) *Reflection*: bouncing back of unabsorbed wave energy when a wave strikes matter.

(xii) *Refraction*: bending of wave due to a change in speed which is a function of the medium of propagation.

(xiii) *Diffraction*: bending of wave around the boundary of a barrier.

(xiv) *Interference*: simultaneous arrival of two or more waves at a place. Same points on interfering waves combine (*Constructive interference*). Opposite points on interfering waves combine (*Destructive interference*). Interfering waves combine at different points, some points add and some points subtract (*Varying interference*).

(xv) *Combination waves*: waves are neither transverse nor longitudinal but a hybrid of traverse and longitudinal waves. The hybrid of transverse and longitudinal mediums movements result in circular movement.

**(b)**

figure 7.7 is a parallel LC *tank* circuit. It is used in frequency applications as a *tuning* or*frequency selection* device.

For a generic parallel LC circuit of figure 7.7, let the resistance = R, the inductance = L, and the capacitance = C

So, for resistor R in series with inductor L, the impedance Z_{1} is:

Z_{1} = R + jωL

For the capacitor C, impedance Z_{2} = 1/jωC

So, admittance Y(ω) = 1/Z_{total} = 1/Z_{1} + 1/Z_{2} = 1/(R + jωL) + jωC-------(1)

So, Y(ω) = R/(R^{2} + (ωL)^{2}) + j[ωC - ωL/(R^{2} + (ωL)^{2})] -------(2)

Equation (2) is the result of multiplying equation (1) with the complex conjugate (R - jωL).

At resonance, ω_{o}C = ω_{o}L/(R^{2} + (ωL)^{2}). Where ω_{o} is the resonant frequency in rad/s.

So, ω_{o} = [(1 - (R^{2}C)/L)^{1/2}]/(LC)^{1/2}-------(3)

Where R is in ohms, C is in Farad, L is in Henry.

Now substituting the values given in figure 7.7 into equation (3)

We have resonant frequency, ω_{o} = 1,000 rad/s = 1,000/2π Hz = 159.2 Hz.

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