Expressions Of Pj Problems

Pj Problems - Overview

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Painting

Frequencies Of Simple Sound And Complex Sound

(a) The sound produced by a tuning fork (figure 113.4) is considered a simple sound. In general, the sound wave of a simple sound can be represented by the following simple sinusoid: **y = asin2πft** --------------(1)

Where *a* is the amplitude of the sound wave, *f* is the frequency and *t* is time.

What is the amplitude and frequency of the simple sound represented by **10sin(π/2)16t**?

(b) Sounds from musical instruments and the human voice are complex sounds so they are not representable by only the simple sinusoid of equation (1). However, intelligible sounds (simple or complex) are periodic even if they are not sinusoids. Consider the following complex sound:
**y = 0.07sin480πt + 0.05sin760πt + ...** -------------(2)

(i) What are the frequencies of the fundamental, first harmonic and first partial?

(ii) What is the frequency of the second harmonic?

(iii) Why is the frequency of a complex sound always that of the first harmonic?

(iv) What is the difference between natural frequency (resonant frequency) and fundamental frequency?

(v) Can humans hear an *infrasonic* or an *ultrasonic* sound?

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

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

(a) General sinusoid of simple sound: **y = asin2πft**

Given sinusoid:**10sin(π/2)16t**?

Comparing both sinusoids:

2πf = (π/2)16

So, f = 8/2 = 4 cycles per second.

Amplitude = 10.

(b)i Fundamental, first harmonic and first partial mean the same thing. *Fourier's theorem says that any periodic function is a sum of simple sine functions representable by equation (1) and that the frequencies of the simple sine functions are all integral multiple of one frequency (the lowest frequency in the series) called the fundamental frequency*.
The simple sine function with the fundamental frequency is called the

So,

So, 480 = 2f; f = fundamental frequency = 240 cycles per sec.

(ii) So, the frequency of the second harmonic is 2(240) = 480 cycles per sec.

So, the frequency of the third harmonic is 3(240) = 720 cycles per sec

Frequencies of subsequent harmonics are similarly calculated. The harmonics beyond the first harmonic are also called

(iii) The cycles of the overtones are contained within the cycle of the fundamental tone.

(iv) All objects have a

(v) Sound waves with frequencies below 20 hertz are called

Range of artificially unaided human hearing is 20 hertz - 20,000 hertz.

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

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