Mathematical Requirements

Although this course is conceptually based, some mathematics is required since the language of mathematics is used by those involved i the technical sciences. To both discuss and do mathematics, we will need a unifying conceptual framework. This framework will be housed in the concept of proportionality. The proportionality will be used in both the lecture and laboratory.

THE PROPORTIONALITY

Mathematical proportionalities are written in the form

In words, A is proportional to B means that if B is doubled, A is also doubled. If B is cut in half, A is also cut in half, etc. Many (but not all) laws of Physics can be written in this form of equation (1). Although equation (1) is fine for determining behaviors, but as stated this form is incomplete. The proportionality symbol can be replaced with an equal sign by introducing a constant,

The variable A and B and the constant are determined by the process under study. Some examples which will be developed further in SCI190 are shown in Table 1.

Table 1. Some typical proportionalities in Physics

STATEMENT	PROPORTIONALITY	LAW	CONSTANT OF PROPORTIONALITY
Motion: The distance of travel of a non-accelerating object is proportional to the travel time	d t	d = v t	Speed (v)
Motion: The acceleration of an object is proportional to the net applied force	a F	a = F/m	Inverse of the mass (1/m)
Energy: The work done by a constant force is proportional to the distance the object moves	W d	W = Fd	Applied force (F)
Waves: The wavelength of a wave is proportional to the period of the wave	T	= v T	Wave Velocity (v)
Electric Circuits: The voltage across an electrical device is proportional to the current through the device	V I	V = RI	Resistance (R)

THE GRAPHICAL INTERPRETATION

The is a convenient graphical interpretation of equation (2). Equation (2) defines a particular version of the equation of a line, often symbolized as

where the slope is defined as

The symbol means change in. We see, therefore, that if a proportionality is plotted, it will produce a straight line with a slope equal to the constant in equation (2).

Notice that the proportionality does not imply that the constant of proportionality cannot be changed. Figure 2 represents the behavior of a moving object at two different (but constant) speeds. Both represent the same proportionality with different constants. As the physical situation changes, the constant of proportionality might also change. This is indicated by the higher slope in Figure 2 (b).

INVERSE PROPORTIONALITIES

A relationship which is similar to the proportionality is the inverse proportionality. In this case,

A 1/B,

or,

A = (constant)/B.

If B is doubled, A is cut in half. If B is tripled, A is reduced by one third. If, on the other hand, B is cut in half, then A is doubled:

A = (constant)/B,

but if B is cut in half, then

(constant)/{B/2} 2 (constant)/B = 2A

since a denominator in a denominator can be moved to the numerator. As a result, A is doubled.

IN THE COURSE

You will learn to work with proportionalities in both lecture and lab, since many of the fundamental laws of physics can be phrased in terms of proportionalities. Good luck!

REVIEW QUESTIONS

1. It is known that A is proportional to B. Information is collected in a laboratory, leading to the following pieces of data: When B = 0, A = 0; when B = 1, A = 4; when B = 2, A = 8; when B = 3, A = 12. What is the value of the constant in the proportionality?

2. It is known that A is proportional to B. Information is collected in a laboratory, leading to the following pieces of data: When B = 0, A = 0; when B = 1, A = 4; when B = 2, A = 8; when B = 3, A = 12. It is also known that A is proportional to C. Information is collected in a laboratory, leading to the following pieces of data: When C = 0, A = 0; when C = 1, A = 2; when C = 2, A = 4; when C = 3, A = 6. Which of the relationships will have the higher slope when graphed, the proportionality between A and B or the proportionality between A and C? Which will have the larger constant in the proportionality?

3. It is known that A is proportional to B. If B is made ten times larger, by how much will A change? If B is made ten times smaller, by how much will A change?

4. It is known that A is inversely proportional to B. If B is made ten times larger, by how much will A change? If B is made ten times smaller, by how much will A change?

5. It is known that A is proportional to B and also inversely proportional to C. If B is made ten times larger, by how much will A change? If B is made ten times smaller, by how much will A change? If C is made ten times larger, by how much will A change? If C is made ten times smaller, by how much will A change? If B is made ten times larger and C is made 5 times larger, by how much will A change? If C is made ten times smaller and B is made 5 times smaller, by how much will A change? If B is made 5 times larger, by how much will C have to change in order for A to remain UNCHANGED? If B is made 5 times smaller, by how much will C have to change in order for A to remain UNCHANGED?

6. The ideal gas law is a proportionality of the form PV T, where P is the pressure, V is the volume, and T is the temperature of the gas. If the Temperature and volume of the gas are both doubled, what happens to the pressure? If the pressure and volume are both doubled, what happens to the temperature?

7. The kinetic energy of an object is proportional to the speed of the object squared. If the speed is doubled, by how much does the kinetic energy change? If the speed is cut in half, by how much does the kinetic energy change?

8. The force between two charges is proportional to the size of each charge, and inversely proportional to the square of the separation between the charges. If the size of each charge is doubled and the separation between the charges is also doubled, by how much will the force between the charges change?