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

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Human-cells

Proteins

Nucleic Acids

COHN - Natures Engineering Of The Human Body

The Human-Body Systems

Vision

Walking

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Sensors Sensings

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Faith, Love, Charity

Photosynthesis

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Systems

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Differential Calculus

Antiderivative

Integral Calculus

Economies

Inflation

Markets

Money Supply

Painting

Incline Planes - The PjProblemStrings

*Inclined planes* and levers are the two main groups of simple machines. Figures 129.1(a) and (b) illustrate a simple inclined plane and a *wedge* respectively.

(a) Explain why the length NQ is longer than the length NM.

(b) Write an expression for the effort that pushes the solid ball into the cart if the weight of the ball is W and the angle of inclination of the plane is θ:

(i) If friction is not at play.

(ii) If friction is taken into account.

(c) The wedge with sides EF = 1" and GH = 3" is used to split a log (figure 129.1(b). How wide does the force of the effort force the log apart?

(d) What is the mechanical advantage of the wedge?

(e) Write the PjProblemstrings of the work done in figures 129.1(a) and 129.1(b).

**The strings**:
S_{7}P_{3}A_{32} (Force-Push).
**The math**:

Pj Problem of Interest is of type *force* (push/pull).

(a) Let NQ = L and NM = *l*.

The work to be done is to lift solid ball into cart. This work is nonvariant.

How the work is done can vary. A very strong person may choose to exert a force F_{1} to lift the solid ball vertically via MN into the cart. A not too strong person with the help of a simple machine such as the inclined plane may exert a lesser force F_{2} to get the solid ball via QN into the cart.

So, work done = F_{1}*l* = F_{2}L .

F_{1}/F_{2} = L/*l* > 1

So, L is longer than *l*.

In order to gain in effort one sacrifices some distance.

(b)i Effort = Wsinθ (neglecting friction)

(b)ii Effort = W(μcosθ + sinθ). Where μ is the coefficient of friction.

(c) The log will be forced apart a distance equal to the width of the broad end of the wedge. This width is 1" for the wedge illustrated in figure 129.1(b).

The wedge is useful in situations where other simple machines are inapplicable.

(d) Mechanical Advantage of wedge , M.A = 3/1 = 3.

(e)Consider figure 129.1(a), if work is done via MN, effort involves *force-pull* to lift solid ball and *force-push* to carry solid ball against the *pull* of gravity into cart in a linear direction.

Assuming a multi-matter-multi-dynamic space (S_{7}) because of the dynamism of atoms of materials and the fact that there are several matter in the space.

So, the PjProblemStrings S_{7}P_{3}A_{31}, S_{7}P_{3}A_{32} and S_{7}P_{4}A_{41} are implied.

If work is done via QN, the inclined plane, no lifting is required

So, only *force-push* is required to push solid ball into cart in a linear direction.

So, S_{7}P_{3}A_{32} and S_{7}P_{4}A_{41} are implied.

Static equilibrium of the plane NQ is required at N and Q.

So, the PjProblemString S_{7}P_{7}A_{71} is implied at N and Q.

Linearly directed *Force-push* resulting in linear motion, at play (effort and resistance) in figure 129.1(b).

So,the PjProblemStrings S_{7}P_{3}A_{32} and S_{7}P_{4}A_{41} are implied.

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