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

Gears - The PjProblemStrings

*Gears* are simple machines. They are used to change the direction of motion; decrease or increase the speed of motion; and magnify or reduce applied force.

(a) Figure 133.1(a) is an illustration of an eggbeater. Gears A, B and C enable the eggbeater to do its work. Physicists say, the *gear* is a type of a lever. Classify the eggbeater as a lever.

(b) Give a range for the theoretical mechanical advantage of the eggbeater of 133.1(a).

(c) Explain how the gears of the eggbeater change the speed of motion.

(d) Figure 133.1(b) is a rack-and-pinion arrangement of gears. What is its usefulness.

(e) Imagine a *gear train* consisting of four gears: A (10 teeth), B (40 teeth) C (20 teeth) and D (10 teeth). A in mesh with B, C is rigidly fixed on the same shaft as B; C in mesh with D. What is the overall *speed ratio* of the gear train?

(f) What is an *idler* gear?

(g) Write the PjProblemStrings at play with respect to figures 133.1(a) and 133.1(b).

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

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

(a) The eggbeater is a *third class lever*. Effort is between resistance and fulcrum and effort applied is greater than resistance inorder to increase speed.

(b) Theoretical Mechanical Advantage (T.M.A) for third class levers is a fraction. So, 0 < T.M.A <1.

(c) Gear A has 32 teeth; gear B has 8 teeth and gear C has 8 teeth.

So, there are four complete revolutions of B to one revolution of A.

So, C also revolves four times since it has the same number of teeth with B.

(d) A *pinion* is the smaller of two gears meshed together. In other words, of the two gears, the *pinion* has fewer teeth. A *rack-and-pinion* gear arrangement is used to change rotary motion to linear motion.

(e) Consider figure 133.2.

Four complete revolutions of A in one complete revolution of B (as a result of respective gear teeth).

B and C fixed to same shaft, so one complete revolution of C in one complete revolution of B.

Two complete revolution of D in one complete revolution of C (as a result of respective gear teeth).

So, overall speed reduction is 1/2.

In general, gear speed formula is as follows:

S_{2} = S_{1}(T_{1}/T_{2}).

Where

S_{1} = speed of first shaft in train

S_{2} = speed of last shaft in train

T_{1} = product of teeth on all drivers

T_{2} = product of teeth on all driven gears

(f) Two external gears that are meshed rotate in opposite directions. An *idler* is a third gear placed between the two gears to make them rotate in the same direction (figure 133.3). The *idler* does not change the gear ratio in speed calculation.

(d) 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.
**Eggbeater (figure 133.1(a)**

Forces at play of type *push*. PjProblemStrings S_{7}P_{3}A_{32}

Motions at play, *rotational* at effort and resistance, PjProblemStrings S_{7}P_{4}A_{42}
**Rack-and-Pinion (figure 133.1(b)**

Forces at play of type *push*. PjProblemStrings S_{7}P_{3}A_{32}

Motion at pinion, *rotational*, PjProblemStrings S_{7}P_{4}A_{42}

Motion at rack, linear, PjProblemStrings S_{7}P_{4}A_{41}

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