Preliminaries - Western Michigan University



Preliminaries

Understanding the basic principles of hearing science does not require an extensive background in mathematics or physics. There are, however, a few basic concepts in mathematical notation and elementary physics that will help pave the way for some of the material that will be encountered later in the text. The two concepts that will be reviewed in this chapter are exponential notation and the representation of physical quantities.

Exponential Notation

In dealing with very large or very small numbers it is often more convenient to express numbers using exponential notation. As will be seen in the next chapter, some familiarity with exponents is needed to understand the decibel scale, which is in widespread use as a measure of sound intensity. The most commonly used exponential notation system is centered around the familiar base 10 number system, although the same idea can be applied to any number system. In all cases, exponential notation involves raising a number called the base to some power, also called the exponent. The exponent indicates how many times the base is used as a factor. Table 2-1 below shows how exponential notation works when using 10 as the base. The numbers in the first column are in "ordinary" notation, the numbers in the center column are in exponential notation, and the factors of 10 that are associated with these numbers are shown in the third column.

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Table 2-1. Exponential notation for even powers of 10.

Exponential

Number Notation Factors of 10

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1 100 (any number raised to the zero power = 1)

10 101 10 x 1 = 10

100 102 10 x 10 = 100

1,000 103 10 x 10 x 10 = 1,000

10,000 104 10 x 10 x 10 x 10 = 10,000

100,000 105 10 x 10 x 10 x 10 x 10 = 100,000

1,000,000 106 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000

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As can be seen in the last column, the exponent simply indicates how many times the number ten is used as a factor. It can also be seen that, for these whole-number powers of ten (i.e., 10 raised to an integer power), the exponent is equal to the number of zeros that are used when the number is written in ordinary notation. The factorizations shown in the last column make intuitive sense, with the possible exception of 100. Although a longer explanation would show that this example makes sense as well, for our purposes it is important simply to note that any number raised to the zero power equals one.

Exponential Notation for Numbers Smaller than One

Positive numbers smaller than one can also be represented using exponential notation. The number 0.1 (1/10) is represented as 10-1 because 0.1 = 1/101. Similarly, 0.01 (1/100) is represented as 10-2 because 0.01 = 1/102. Using the same logic, 10-5 = 1/105 = 1/100,000 = 0.00001. In all cases, 10-x = 1/10x. Table 2-2 below shows how the system works. Notice that for these even powers of ten, the exponent (with a negative sign) is equal the number of zeros that appear after the decimal point plus one; e.g., for 0.0001, there are 3 zeros after the decimal point, giving an exponent of –(3+1) = -4.

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Table 2-2. Exponential notation for numbers less than 1.

Exponential

Number Notation Factors of 10

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0.1 10-1 1/(10x1) = 1/101

0.01 10-2 1/(10x10) = 1/102

0.001 10-3 1/(10x10x10) = 1/103

0.0001 10-4 1/(10x10x10x10) = 1/104

0.00001 10-5 1/(10x10x10x10x10) = 1/105

0.000001 10-6 1/(10x10x10x10x10x10) = 1/106

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Logarithms

In a number such as 102, the number 10 is referred to as the base, and the number 2 can be referred to either as a power, an exponent, or a logarithm. These three terms – power, exponent, and logarithm – are completely interchangeable. When you take the logarithm of a number, you are simply finding the exponent that is used when the number is written in exponential notation. For example, the base-10 logarithm (log10 for short) of 100 is 2 because 100 = 102. Similarly:

log10 1,000 = 3 (because 1,000 = 103)

log10 1,000,000 = 6 (because 1,000,000 = 106)

log10 1 = 0 (because 1 = 100)

log10 0.1 = -1 (because 0.1 = 10-1)

log10 0.0001 = -4 (because 0.0001 = 10-4)

All of the examples above show logarithms for whole-number powers of ten such as 10, 100, 1,000, etc. However, logarithms can be found for any positive number. For example, it stands to reason that the logarithm of 200 has to be somewhere between 2 and 3 because the number 200 is greater than 100 (which has a log10 of 2) but less than 1,000 (which has a log10 of 3). If you have a calculator with a log button, you can confirm that the log10 of 200 is approximately 2.3, which means that 102.3 = 200. There are several methods that can be used to obtain the logarithm of a number that is not a whole number power of ten, but the simplest by far is to use a hand calculator that is equipped with a log function.

Arithmetic Using Exponential Notation

We have seen that exponential notation provides a convenient method for representing very large or very small numbers (e.g., 109 instead of 1,000,000,000). An additional advantage is that certain arithmetic operations become very simple when numbers are expressed in exponential notation. For example:

1. Multiplication (add exponents)

102 x 103 = 105

102 x 10-5 = 10-3

2. Division (subtract exponents)

105 ÷ 103 = 102

107 ÷ 10-2 = 109

3. Raising a number to a power (multiply exponents)

(103)2 = 106 (103 raised to the second power = 106)

(105)3 = 1015 (105 raised to the third power = 1015)

Representation of Physical Quantities

As we will see soon, understanding how sound is produced and measured requires some understanding of physical units such as force, pressure, intensity, etc. Our understanding of these physical quantities need not be very detailed, but a general understanding will make it easier to grasp certain concepts that are introduced later on.

Basic Versus Derived Properties

There are many quantities that are used in physics, but only three of these are considered to be basic or fundamental. The three fundamental quantities are:

• distance (also length)

• mass

• time

All other quantities, such as velocity, acceleration, work, force, pressure, etc., are derived by combining these basic quantities in various ways. For example, velocity is simply the change in the position of an object (a measure of distance) divided by the time taken to travel that distance (e.g., 60 miles per hour, 300 meters per second, etc.). So, we got the derived property of velocity by combining the basic properties of distance and time. There are several derived properties that are needed to understand the production and measurement of sound.

Velocity and Acceleration

As mentioned above, velocity is the change in the position of an object (Δx) divided by the time over which that position change occurred:

v = Δx/t, where: v = velocity (more precisely, average velocity)

Δx = change in position (the Δ symbol means “change in”)

t = time over which the position change occurred

Acceleration is the change in velocity over time. You have probably heard it said that a certain sports car can do "0 to 60 in 6 seconds." The "0 to 60 in 6 seconds" is a measure of acceleration: the change in velocity is 60 mph and the time over which the change occurred is 6 seconds. The formula looks like this:

a = Δv/t, where: a = acceleration (more precisely, average acceleration)

Δv = the change in velocity

t = time over which velocity change occurred

The formula may make acceleration look more obscure than it is. Acceleration is simply the change in velocity divided by the time over which the velocity change occurred. In the example above, the change in velocity is 60 mph and the time over which the velocity change occurred is 6 s, so the acceleration is 10 mph/s. The example above is an instance of speeding up, or positive acceleration; slamming the brakes on would cause the car to slow down. This can be referred to either as negative acceleration or deceleration.

Force

Imagine a billiard ball sitting motionless on a perfectly level pool table. You know from every day experience – and from Newton's first law of motion that you probably learned in seventh-grade science class – that the ball will remain motionless unless acted upon by an outside influence. What is needed to move the ball is a force. Force can be thought of as the amount of "push" or "pull" on an object. Force is the mass of the object multiplied by its acceleration:

F = Ma, where: F = force

M = mass

a = acceleration

The formula tells us that force can be increased either by increasing mass or by increasing acceleration (or, of course, both). So, our billiard ball will be highly disturbed if it is struck by a massive, highly accelerated object. On the other hand, the ball will be disturbed by a small amount if it is struck by a light object that is moving with a low rate of acceleration. Notice that it is acceleration and not velocity that is a determinant of force. The old saying, ‘it is not the fall that hurts, it is the sudden stop at the end’ expresses exactly this fact about the role acceleration in generating force. No force is generated by an object that is traveling at a constant velocity (and, therefore, zero acceleration), no matter how great the velocity. Force requires a change in velocity; that is, force requires acceleration. This explains why airline passengers feel the force of the airplane primarily during takeoff and landing but not in the middle of the flight when the plane is cruising at a high but relatively constant velocity. This also explains why we have no sensation of the exceedingly high velocity with which the earth is simultaneously revolving and its axis and hurtling through space.

Pressure

As will be discussed in later chapters, the distinction between force and pressure is quite important for understanding how the middle ear works. Force, as we have seen, is the amount of push or pull on an object. Pressure, on the other hand, is force per unit area, or the amount of force that is "focused" on a given area. The formula is:

E = F/A, where: E = pressure[1]

F = force

A = area over which force is delivered

The difference between force and pressure can be illustrated with the objects shown in Figure 2-1, which we can assume are identical in mass and differ only with respect to orientation. If each of these objects were dropped on a napkin from the same height, they would strike the napkin with exactly the same force since both mass and acceleration would be identical (F = Ma). It is obvious that the object on the right would be more likely to tear the napkin, and the reason is that the force is being delivered over a much smaller area, producing much greater pressure on impact. So, in the object on the right, pressure has been amplified without an increase in force by decreasing the area over which the force is delivered. Exactly the same principle explains: (1) why a sharp knife cuts better than a dull knife, (2) why thumb tacks are inserted into a bulletin board pointy-end first rather than vice versa, and (3) why a woman would be more likely to sink into a wet lawn if she is wearing stiletto heels rather than tennis shoes. As we will see later, the middle ear exploits this principle to amplify the pressure wave that is delivered to the inner ear.

Work

Work is accomplished when a force that is applied to an object causes the object to move. The amount of work that is done increases when the size of the force increases. Also, more work is done if the object is moved a greater distance. The formula is:

W = Fx, where: W = work

F = force

x = distance

The formula indicates that a large amount of work is done when large forces cause objects to move great distances. The principle can be illustrated with a simple example. Imagine that one laborer moves 100 kilograms of bricks a distance of 30 meters, and a second laborer applies an equal force to move the same 100 kilograms of bricks a distance of 60 meters. Both the formula and intuition tell us that the second laborer has accomplished twice as much work as the first laborer.

Power

Power is just one step more complicated than work. Power is work per unit time, or the rate at which work is accomplished. The formula is:

P = W/t, where: P = power

W = work

t = time taken to complete the work

Power basically tells us how rapidly work is being accomplished. We can return to our example of the two laborers. Suppose that two laborers apply identical forces to move 100 kilograms of bricks a distance of 30 meters. Since both the force and the distance are identical, the two laborers have done the same amount of work (W = Fx). However, suppose that one laborer completes the job in 30 minutes while the other completes the same job in just 15 minutes. Since power is the rate at which work is accomplished, twice as much power was generated by the second laborer than the first.

Intensity

Intensity, which is one step more complicated than power, is simply power per unit area; that is, intensity is the amount of power that is "focused" on a given area. The formula is:

I = P/A, where: I = intensity

P = power

A = area over which the power is measured

Notice that the relationship between power and intensity is analogous to the relationship we saw earlier between force and pressure; that is, pressure is force per unit area, and intensity is power per unit area. There is, in fact, a close relationship between intensity and pressure, and this relationship is important for understanding how the decibel scale works.

Table 2-3 presents a summary of the physical quantities discussed above, along with a non-exhaustive list of some of the units that are used to measure quantities such as acceleration, pressure, work, and so on.

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Table 2-3. Summary of physical quantities.

Quantity Formula Units Concept

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velocity v= Δx/t meters per sec high velocity means large changes in

miles per hour position occurring in a short period of time

Δx= change in position

t=time

acceleration a = Δv/t meters/sec squared speeding up or slowing down; i.e.,

miles per hour per second a change in velocity over time

Δv = change in vel.

t = time

force F = Ma Newton amount of "push" or "pull" on an object

dyne

M = mass

a = acceleration

pressure E = F/A Newton/sq meter pressure is high when a large force

dyne/sq centimeter is delivered over a small area

F = force Pascal

A = area

work W = Fx joule a large amount of work is done when a large

erg force moves an object a great distance

F = force foot-pound

x = distance

power P = W/t watt the rate at which work is done

horsepower

W = work

t = time

intensity I = P/A watt/square meter intensity is high when a large amount of

watt/square centimeter power is focused on a small area

P = power

A = area

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1. In the expression 104, the number 10 is called the _________. Three different terms can be used to refer to the number 4 in this expression: ____________, ____________, or ____________.

2. Any number raised to the zero power equals: ___________.

3. 10-7 is equivalent to 1 divided by _________. Written in ordinary, non-exponential notation, this is equal to _________.

4. Since 105 = 100,000, the log10 of 100,000 = _________.

5. 108 / 102 = ______. Written in ordinary, non-exponential notation, this is equal to _________.

6. 104 / 109 = ______. Written in ordinary, non-exponential notation, this is equal to _________.

7. 105 x 103 = ______. Written in ordinary, non-exponential notation, this is equal to _________.

8. 10-3 = ________.

9. (102)3 = ______. Written in ordinary, non-exponential notation, this is equal to _________.

10. The log10 of 435,973 is greater than ______ but less than ______.

11. These three physical quantities are considered to be basic or fundamental: ____________, ____________, and ____________. Quantities that are arrived at by combining these basic quantities in various ways are referred to as ____________ quantities.

12. Supply the physical quantity for each of the descriptions below.

a. Amount of push or pull on an object (or the product of mass and acceleration): ____________.

b. Distance traveled divided by the time required to travel that distance: ____________.

c. The product of force and distance: ____________.

d. The rate at which work is accomplished: ____________.

e. Power per unit area: ____________.

f. Force per unit area: ____________.

g. Change in velocity over time: ____________.

13. Laborer A moves 50 kilograms of cottage cheese a distance of 2 kilometers in 30 minutes. Laborer B moves the same 50 kilograms the same distance but takes 60 minutes to finish the job. Measured in terms of ____________ the two laborers are therefore identical. However, because of the difference in the amount of time required to complete the job, measures of ____________ will be twice as large for laborer A than laborer B.

14. The difference between force and pressure is analogous to the difference between power and ____________.

15. The objects in Figure 2-1 are dropped from the same height. When they strike the surface below, ____________ will be identical for the two objects but ____________ will be greater for the object with the pointy tip facing down.

16. The purpose of sharpening a knife is to:

a. amplify force

b. amplify pressure

c. amplify acceleration

d. decrease velocity

e. all of the above

f. a and b only

g. a, b, and c

Study Question Answers: Preliminaries

1. In the expression 104, the number 10 is called the base. Three different terms can be used to refer to the number 4 in this expression: power, exponent, or logarithm.

2. Any number raised to the zero power equals: 1.

3. 10-7 is equivalent to 1 divided by 107. Written in ordinary, non-exponential notation, this is equal to 0.0000001.

4. Since 105 = 100,000, the log10 of 100,000 = 5.

5. 108 / 102 = 106. Written in ordinary, non-exponential notation, this is equal to 1,000,000.

6. 104 / 109 = 10-5. Written in ordinary, non-exponential notation, this is equal to 0.00001.

7. 105 x 103 = 108. Written in ordinary, non-exponential notation, this is equal to 100,000,000.

8. 10-3 = 1/103 or 0.001

9. (102)3 = 106. Written in ordinary, non-exponential notation, this is equal to 1,000,000.

10. The log10 of 435,973 is greater than 5 but less than 6.

11. These three physical quantities are considered to be basic or fundamental: time, mass, distance. Quantities that are arrived at by combining these basic quantities in various ways are referred to as derived quantities.

*** See text for answers to the questions below ***.

12. Supply the physical quantity for each of the descriptions below:

a. Amount of push or pull on an object (or the product of mass and acceleration): ____________.

b. Distance traveled divided by the time required to travel that distance: ____________.

c. The product of force and distance: ____________.

d. The rate at which work is accomplished: ____________.

e. Power per unit area: ____________.

f. Force per unit area: ____________.

g. Change in velocity over time: ____________.

13. Laborer A moves 50 kilograms of cottage cheese a distance of 2 kilometers in 30 minutes. Laborer B moves the same 50 kilograms the same distance but takes 60 minutes to finish the job. Measured in terms of ____________ the two laborers are therefore identical. However, because of the difference in the amount of time required to complete the job, measures of ____________ will be twice as large for laborer A than laborer B.

14. The difference between force and pressure is analogous to the difference between power and ____________.

15. The objects in Figure 2-1 are dropped from the same height. When they strike the surface below, ____________ will be identical for the two objects but ____________ will be greater for the object with the pointy tip facing down.

16. The purpose of sharpening a knife is to:

a. amplify force

b. amplify pressure

c. amplify acceleration

d. decrease velocity

e. all of the above

f. a and b only

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[1]The symbol 'E' will be used for pressure because 'P' will be needed later for power.

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Figure 2-1. Two objects with identical masses that differ only in orientation. If the two objects were dropped from the same height, they would strike a surface below with identical force. However, the pressure generated by the object on the right would be much greater because the area over which the force is delivered would be considerably smaller; that is, the pressure would be greater.

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