Measurements and Units

A large part of science is based upon the creation of mathematical models that describe physical phenomena. For example, the subject of Newtonian Mechanics describes how objects will behave when they are subject to forces. We can use Newtonian Mechanics to model our solar system allowing us to accurately predict the positions of the sun and the planets. However, to relate the mathematical description to the real world requires that there is a mutually agreed upon measurement system so that, for example, the distance between the earth and the sun is assigned the same numerical value by different scientists in different parts of the world. To accomplish this task we use a mutually agreed upon system of units.

Currently, there are actually two widely used systems of units. The first-British Engineering Units-is, strangely enough, used primarily in the USA and almost no place else (certainly not in Britain!). The second is the Metric System (or System International [SI]) which is an internationally agreed upon standard. We will mostly use the metric system for problems in this class.

The Standard Prefixes used in SI Units

One nice thing about the SI system is that there is a uniform prefix system that is used across all measured quantities. For example the prefix kilo means 1000. Thus a kilogram is 1000 grams or a kilometer is 1000 meters. All of the prefixes are some multiple of 10; thus, there is no remembering awkward quantities such as 12 inchs in one foot, three feet in a yard, 5280 feet in a mile etc. You should memorize the following prefixes (I actually hope that you pretty much know most of them already!). You should also be able to manipulate quantities in scientific notation. Try the problems at the bottom of this page to see if you are up on working with numbers in scientific notation.

Prefix Multiplier
femto 10-15
nano 10-9
micro 10-6
milli 10-3
centi 10-2
deca 10
kilo 103
mega 106
giga 109
tera 1012

The Basic Units: Meter, Kilogram, and Second

Most of the quantities that we will discuss this semester have units that are expressable as some combination of the three basic quantities Meter, Kilogram, Second which measure length, mass, and time respectively.

UNIT SYMBOL DESCRIPTION
Meter m Measures length
Kilogram kg Measures mass (different than weight!)
Second s Measures time

Vector and Scalar Quantities

There is one more wrinkle to the whole question of measured quantities. Some quantities are not sufficiently expressed by a mere number, they require the additional information of a direction. Quantities which require a direction to fully describe them are called vectors. Quantities that are described merely by a number are called scalars. The difference between vectors and scalars is best understood by an example: the distance (a scalar) from Murfreesboro to Nashville is 58 kilometers whereas the displacement (a vector that points from M'boro to N'ville) is 58 kilometers in a northwest direction. Note that the vector information is considerably more useful if you are planning to hike between the two places!

UNIT SYMBOL VECTOR DESCRIPTION
Displacement m Y Vector that connects two points in space.
Speed m s-1 N Rate of change of position
Velocity m s-1 Y Rate of change of displacement
Acceleration m s-2 Y Rate of change of velocity
Force N = kg m s-2 Y Push or pull (see below for more info)
Pressure N m-2 N Force per unit area
Work/Energy J = N m N Work=force x distance, energy (see below)
Power W = J s-1 N Energy expended or work done per second

 

Note that in the above table N stands for Newton, the unit of force; J stands for Joule the unit of work or energy; and W stands for Watt the unit of power. Even though these quantities have more elaborate names they are nonetheless composed of our basic units the meter, kilogram, and second.

In class we will more fully discuss the nature of forces. The basic concept that will be developed is that any object that experiences a net force will accelerate. The relation between the net force F, the objects mass m, and the resulting acceleration a is described by Newton's Second Law of motion F = ma.

Similarly we will further amplify the concepts of work and energy. The table above is just a simplified reminder of what we do in class.

Problems

After class you should be able to attempt the following questions and problems. These exercises are representative of the sort of problems that will be on the quiz. The answers are given at the bottom of the page. If you do not understand how to do a problem even after you look at the answer you can ask me after class or in my office hours.

Scientific notation problems

  1. Express the following numbers in scientific notation:

    1. 0.012 (b) 0.0000005 (c) 52500 (d) 65

  2. Convert the following numbers from scientific notation to regular decimal numbers:

    1. 3.5 x 10-4 (b) 2.5 x 106 (c) 8 x 10-12 (d) 100.

  3. Perform the following mathematical operations leaving your answers in scientific notation:

    1. 102 + 103

    2. 102 + 102

    3. 102 x 10-2

    4. 106 x 104

    5. 106/104

    6. 3 x 103 x 2 x 108

    7. 3 x 10-2 x 2 x 10-5

  4. Multiply 3.73 x 102 by 4.44 x 103 and write the resulting number in scientific notation. Note that whereas the numbers 3.73 and 4.44 are multiplied together the superscripts on the 10’s add to give the final result. This feature of “multiplying by adding” will crop up later in the semester when we look at the mathematics of logarithms.

  5. A car travels at a constant speed of 32 ms-1. How many kilometers does it travel in 10 minutes?

  6. You walk 10m North, turn and walk 25 m South. What distance do you travel? What is your displacement with respect to your starting point? If it takes you 10 seconds to make the entire trip, what is your speed? (assume that you moved the whole time at a constant speed).

  7. How many nanoseconds are there in a day?

  8. You are 2 km away from a lightning strike. How long does it take the light of the lightning flash to reach you? How long does it take for the sound (the thunder) to reach you? The speed of light can be taken to be 3x108 m/s and the speed of sound to be 340 m/s.

  9. You throw a ball straight up into the air. At the instant the ball leaves your hand, what force(s) act on the ball? What is the ball's acceleration? At the very peak of the ball's motion its velocity is zero for an instant, what is its acceleration at this point?

  10. The average outer surface area of a human body is about 2 m2. Calculate the total force of atmospheric air pressure on the outer surface of the body. Remember in MKS units PA=105 Nm-2.

  11. What is the approximate volume of air in a shoe box? What is the surface area of the outside of a shoe box? What are the units of volume and area? Is volume a vector or a scalar quantity? [Knowing how to calculate the volume and surface area of simple geometric shapes become important when we look at auditorium acoustics later in the semester.]

The Answers (don't peek before you try the questions now!)

  1. (a) 1.2 x 10-2 (b) 5 x 10-7 (c) 5.25 x 104 (d) 6.5 x 101 = 6.5 x 10
  2. (a) 0.00035 (b) 2,500,000 (c) 0.000000000008 (d) 1
  3. (a) 1.1 x 103 (b) 2 x 102 (c) 1 (d) 1010 (e) 102 (f) 6 x 1011 (g) 6 x 10-7
  4. 16.56 x 105 = 1.656 x 106
  5. 32 x 10 x 60 = 19200 m = 19.2 km
  6. 35 m is the total distance traveled. Displacement is 15 m South (number and direction because displacement is a vector!). Speed= 35m/10s = 3.5 ms-1.
  7. 24 hr/day x 60 min/hr x 60 s/min x 109 ns /s = 8.64 x 1013 ns/day.
  8. Speed = distance traveled / time which can be rearranged to be time = distance / speed. For sound the time is 2000/340 = 5.88 s. For light the time is 2000/3 x 108 = 6.67 x 10-6.
  9. While the ball is in the air the only force that acts on it is the force of gravity. Because the force of gravity is given by mg (m is the mass of the ball and g is 9.8 ms-2), Newtons second law tells us that the acceleration of the ball is = 9.8 ms-2 at all times during the flight of the ball even at the top when it stops for an instant.
  10. P=F/A therefore total force F=PA=105 Nm-2 x 2 m2 = 2 x 105 N.
  11. The width, height, and length of a shoe box are approximately W=20 cms, H=15 cms, and L=35 cms respectively (your approximate values may be slightly different). The volume (width x height x length) is given by WxHxL=20x15x35 cm3=10500 cm3. The surface area is given by adding up the surface area of the 6 sides of the box. With my numbers I get 3050 cm2. Volume is a scalar--after all, what direction would a volume point?

Lastly, here is a javascript conversion utility that I copied off the web. It's hard to read. It's just something you might find useful.

Length Equivalents
First, type the number you wish converted here:
Then, click radio buttons for desired conversion:
From: Centimeters
 Inches
 Feet
 Yards
 Meters
 Chains
 Kilometers
 Miles
To: Centimeters
 Inches
 Feet
 Yards
 Meters
 Chains
 Kilometers
 Miles

AAPT2004 Audio Demos & Slides

Send me an email at wroberts@mtsu.edu

Contact Information

Dr. W. M. Robertson
MTSU Box X-116
Murfreesboro, TN 37132
Ph. (615) 898-5837