Measurements are a basic necessity in science. Scientists have designed thousands of scales, rulers and other measuring tools to help in the vital process of measuring. In this image of the control panel of the space shuttle Atlantis, we see dozens of readouts from measuring systems.
Measurement
Observation is an integral part of the scientific method. Hypotheses are accepted or rejected based on how well they explain observations. Some observations have numbers associated with them and some do not. An observation such as “the plant turned brown” is called a qualitative observation because it does not have any numbers associated with it. An observation such as “the object moved 200 meters” is called a quantitative observation because it contains a number. Quantitative observations are also called measurements. The numerical component of the observation is obtained by measurement, i.e. comparing the observation to some standard even if the comparison is an estimate. In terms of value to a scientist, all observations are useful but quantitative observations are much more useful. Whenever possible, you should make quantitative rather than qualitative observations, even if the measurement is an estimate.
Since accurate measurement is a vital tool in performing science, it becomes obvious that a consistent set of units for measurement is necessary. Physicists throughout the world use the Internation System of Units (also called the SI system). The SI system is basically the metric system, which is a convenient measuring system because units of different size are related by powers of 10. The system has physical standards for length (m), mass (kg), and time (s). These are called fundamental units because they have an actual physical standard. There are four additional fundamental units that measure electric current (A), temperature (K), luminous intensity (cd) and the amount of a substance (mol).
The standard SI unit for length is the meter. Originally, the definition of the meter was the distance between two scratches on a length of metal. The standard length of metal was stored in a secure vault under controlled conditions of temperature, pressure, and humidity. Most countries had their own copies of the standard meter and many copies were made for actual use. Later, the standard was redefined as one ten-millionth of the distance from the north pole to the equator measured along a line that passed through Lyons, France. In 1960, the standard was redefined again as a multiple of a wavelength of light emitted by krypton-86. In 1982, the standard was redefined yet again as the distance light travels in 1/299792458 second in a vacuum.
The standard unit of time, the second, was once defined as a fraction of the time it takes the earth to complete it orbit around the sun but has now been redefined in terms of the frequency of one type of radiation emitted by a cesium-133 atom.
The standard unit for mass is the kilogram. This standard is a mass of platinum-iridium metal cylinder kept near Paris, France. Other countries, of course, keep copies.
Units that are expressed using combinations of fundamental units are called derived units. For example, length is a fundamental unit measured in meters, time is a fundamental unit measured in seconds, and speed is a derived unit measured in meters/second.
Some common types of measurements and their symbols used in physics is summarized in the table below.
Type of measurement | Commonly used symbols | Fundamental units |
---|---|---|
length or position |
\begin{align*} d, x, L \end{align*} |
meters (m) |
time |
\begin{align*} t \end{align*} |
seconds (s) |
velocity or speed |
\begin{align*} v, u \end{align*} |
meters per second (m/s) |
mass |
\begin{align*} m \end{align*} |
kilograms (kg) |
force | \begin{align*}\mathbf{F} \end{align*} | Newtons (N) |
energy | \begin{align*} E, K, U, Q \end{align*} | Joules (J) |
power | \begin{align*} P \end{align*} | Watts (W) |
electric charge | \begin{align*} q, e \end{align*} | Coulombs (C) |
temperature | \begin{align*} T \end{align*} | Kelvin (K) |
electric current | \begin{align*} I \end{align*} | Amperes (A) |
electric field | \begin{align*}\mathbf{E} \end{align*} | Newtons per Coulomb (N/C) |
magnetic field | \begin{align*}\mathbf{B} \end{align*} | Tesla (T) |
As mentioned earlier, the SI system is a decimal system. Prefixes are used to change SI units by powers of ten. Thus, one hundredth of a meter is a centimeter and one thousandth of a gram is a milligram. The metric units for all quantities use the same prefixes. One thousand meters is a kilometer and one thousand grams is a kilogram. The common prefixes are shown in the table below.
Prefix | Symbol | Fractions | Example |
pico | p | 1 × 10^{-12} | picometer (pm) |
nano | n | 1 × 10^{-9} | nanometer (nm) |
micro | μ | 1 × 10^{-6} | micgrogram (μg) |
milli | m | 1 × 10^{-3} | milligram (mg) |
centi | c | 1 × 10^{-2} | centimeter (cm) |
deci | d | 1 × 10^{-1} | decimeter (dm) |
Multiples | |||
tera | T | 1 × 10^{12} | terameter (Tm) |
giga | G | 1 × 10^{9} | gigameter (Gm) |
mega | M | 1 × 10^{6} | megagram (Mg) |
kilo | k | 1 × 10^{3} | kilogram (kg) |
hecto | h | 1 × 10^{2} | hectogram (hg) |
deka | da | 1 × 10^{1} | dekagram (dag) |
* Please note: you will need to put to memory those prefixes above that are highlighted in blue *
Equivalent measurements with different units can be shown as equalities such as 1 meter = 100 centimeters. Each of the prefixes with each of the quantities has equivalency statements. For example, 1 gigameter = 1 × 10^{9} meters and 1 kilogram = 1000 grams. These equivalencies are used as conversion factors when units need to be converted.
You probably have noticed a few odd looking symbols in the tables above. Many of these symbols are Greek letters and are frequently used in physics. The table below highlights some of the commonly used Greek letters that are used as symbols in physics.
\begin{align*} \mu \end{align*} “mu” | \begin{align*} \tau \end{align*} “tau” | \begin{align*} \Phi \end{align*} “Phi”^{*} | \begin{align*} \omega \end{align*} “omega” | \begin{align*} \rho \end{align*} “rho” |
---|---|---|---|---|
\begin{align*} \theta \end{align*} “theta” | \begin{align*} \pi \end{align*} “pi ” | \begin{align*} \Omega \end{align*} “Omega”^{*} | \begin{align*} \lambda \end{align*} “lambda” | \begin{align*} \Sigma \end{align*} “Sigma”^{*} |
\begin{align*} \alpha \end{align*} “alpha” | \begin{align*} \beta \end{align*} “beta” | \begin{align*} \gamma \end{align*} “gamma” | \begin{align*} \Delta \end{align*} “Delta”^{*} | \begin{align*} \epsilon \end{align*} “epsilon” |
Two very common Greek letters are \begin{align*}\Delta\end{align*} and \begin{align*}\Sigma\end{align*} . \begin{align*}\Delta\end{align*} is used to indicate that we should use the change or difference between the final and initial values of that specific variable. \begin{align*}\Sigma\end{align*} denotes the sum or net value of a variable.
- Every answer to a physics problem must include units. Even if a problem explicitly asks for a speed in meters per second (m/s), the answer is 5 m/s, not 5.
- If a unit is named after a person, it is capitalized. So you write “10 Newtons,” or “10 N,” but “10 meters,” or “10 m.”
- Metric units use a base numbering system of 10. Thus a centimeter is ten times larger than a millimeter. A decimeter is 10 times larger than a centimeter and a meter is 10 times larger than a decimeter. Thus a meter is 100 times larger than a centimeter and 1000 times larger than a millimeter. Going the other way, one can say that there are 100 cm contained in a meter.
Example 1
Question: Convert 2500 m/s into km/s
Solution: A km (kilometer) is 1000 times bigger than a meter. Thus, one simply divides by 1000 and arrives at 2.5 km/s
Example 2
Question: The lengths of the sides of a cube are doubling each second. At what rate is the volume increasing?
Solution:The cube side length, \begin{align*}x\end{align*}, is doubling every second. Therefore after 1 second it becomes \begin{align*}2{x}\end{align*}. The volume of the first cube of side \begin{align*}{x}\end{align*} is \begin{align*}{x}\times{x}\times{x}={{x}^3}\end{align*}. The volume of the second cube of side \begin{align*}2{x}\end{align*} is \begin{align*}2{x}\times2{x}\times2{x}={8{x}^3}\end{align*}. The ratio of the second volume to the first volume is \begin{align*}{8{x}^3}/{{x}^3}=8\end{align*}. Thus the volume is increasing by a factor of 8 every second.
Example 3
Question: Convert 500. millimeters to meters.
Solution: The equivalency statement for millimeters and meters is 1000 mm = 1 m. To convert 500. mm to m, we multiply 500. mm by a conversion factor that will cancel the millimeter units and generate the meter units. This requires that the conversion factor has meters in the numerator and millimeters in the denominator.
Example 4
Question: Convert 11 µg to mg.
Solution: We need two equivalency statements because we need two conversion factors.
\begin{align*}1 \times 10^9 \ \mu \text{g} = 1 \ \text{g and} \ 1000 \ \text{mg} = 1 \ \text{g} \end{align*}
\begin{align*}(11 \ \mu \text{g}) \left(\frac{1 \ \text{g}}{1 \times 10^{-9} \ \mu \text{g}}\right) \left(\frac{1000 \ \text{mg}}{1 \ \text{g}}\right)=1.1 \times 10^{-5} \ \text{mg}\end{align*}
The first conversion factor converts from micrograms to grams and the second conversion factor converts from grams to milligrams.
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Time for Practice
- A tortoise travels \begin{align*}15\end{align*} meters \begin{align*}\;\mathrm{(m)}\end{align*} west, then another \begin{align*}13\end{align*} centimeters \begin{align*}\;\mathrm{(cm)}\end{align*} west. How many meters total has she walked?
- A tortoise, Bernard, starting at point A travels \begin{align*}12 \;\mathrm{m}\end{align*} west and then \begin{align*}150\end{align*} millimeters \begin{align*}\;\mathrm{(mm)}\end{align*} east. How far west of point \begin{align*}A\end{align*} is Bernard after completing these two motions?
- \begin{align*}80 \;\mathrm{m} + 145 \;\mathrm{cm} + 7850 \;\mathrm{mm} = X\ \;\mathrm{mm} \end{align*}. What is\begin{align*} X \end{align*} ?
- A square has sides of length \begin{align*}45 \;\mathrm{mm}\end{align*}. What is the area of the square in \begin{align*}\;\mathrm{mm}^2\end{align*}?
- A square with area \begin{align*}49 \;\mathrm{cm}^2\end{align*} is stretched so that each side is now twice as long. What is the area of the square now? Include a sketch.
- A rectangular solid has a square face with sides \begin{align*}5 \;\mathrm{cm}\end{align*} in length, and a length of \begin{align*}10 \;\mathrm{cm}\end{align*}. What is the volume of the solid in \begin{align*}\;\mathrm{cm}^3\end{align*}? Sketch the object, including the dimensions in your sketch.
- As you know, a cube with each side \begin{align*}4 \;\mathrm{m}\end{align*} in length has a volume of \begin{align*}64 \;\mathrm{m}^3\end{align*}. Each side of the cube is now doubled in length. What is the ratio of the new volume to the old volume? Why is this ratio not simply \begin{align*}2\end{align*}? Include a sketch with dimensions.
- What is the ratio of the mass of the Earth to the mass of a single proton? (Look up the masses online.)
- A spacecraft can travel \begin{align*}20 \;\mathrm{km/s}\end{align*}. How many km can this spacecraft travel in 1 hour \begin{align*}\;\mathrm{(h)}\end{align*}?
- Convert 76.2 kilometers to meters.
- Convert 76.2 picograms to kilograms.
- Convert 1 day into seconds.
Answers
- \begin{align*}15.13 \;\mathrm{m}\end{align*}
- \begin{align*}11.85 \;\mathrm{m}\end{align*}
- \begin{align*}89,300 \;\mathrm{mm}\end{align*}
- \begin{align*}2025 \;\mathrm{mm}^2\end{align*}
- \begin{align*}196 \;\mathrm{cm}^2\end{align*}
- \begin{align*} 250 \;\mathrm{cm}^3\end{align*}
- \begin{align*}8:1,\end{align*} each side goes up by \begin{align*}2 \;\mathrm{cm}\end{align*}, so it will change by \begin{align*}2^3\end{align*}
- \begin{align*}3.5 \times 10^{51}:1\end{align*}
- \begin{align*} 72,000 \;\mathrm{km/h}\end{align*}
- 76,200 m
- 7.62 x 10^{-14} kg
- 86,400 s
Summary
- Measurements (quantitative observations) are more useful than qualitative observations.
- You should make measurements, even estimated ones, whenever possible.
- The system of units for measurements in physics is the SI system.
- The fundamental quantities we will be studying are length, mass, and time.
- Prefixes are used to change SI units by powers of ten.
- Equivalencies are used as conversion factors when units need to be converted.