Working with Values from Measurement
Physics is a science based on measurements. Today, we will talk about how
physicists report and manipulate their measurements. This requires
understanding of the following
As I said in the first lecture, physics is "built" upon basics. Understanding
measured quantities and being able to manipulate them is one such pillar at
the foundation of physics. You may want to use the hyperlinks in the above
list to "jump" over detail about things you are already comfortable with.
Physicists distinguish two types of quantities. The first are exact
such as the number \pi=3.14159..., rational fractions (a number that can be
written as the ratio of two whole numbers), etc. These numbers have precise
definitions, carry no units and can be specified to arbitrary precision.
These are the kinds of numbers you are most familiar with.
The second kind of quantity is the result of a measurement. Measured
quantities have the following characteristics
- The measurement value has a precision determined by...
- the number of digits used in writing the number. The most common
way of writing numbers in this fashion is to use scientific notation . Let's
first see how the number of significant digits implies the
accuracy of the measurement. For example,
the time it took Joe to walk from Swain to his apartment
was measured to be 35 seconds. By using two significant
digits, we imply that the time is uncertain to within ±1
second. The true time for Joe's walk was somewhere between
34 and 36 seconds. Quoting the measurement as 35 seconds is
different from quoting it as 35.0 seconds, the latter implies an
uncertainty of ±0.1 second
- specific reference to the measurement uncertainty. For example,
the length of a pendulum string is measured to be 50±2
cm. This means that the true length of the pendulum string is
between 48 and 52 cm.
- The measurement value has units, discussed below.
It is important to always keep in mind these two features of measured
quantities. When calculating new quantities from measurements, it is a common
error to use all of the digits your calculator has, falsely implying that the
deduced quantity is very precise. As well, be sure to always include the
measurement units!
Because of the implied precision, reporting measurements is tricky and requires
a different way of writing numbers. For example, what happens if Jim takes a
walk measured to last for 350 seconds, but the measurement is accurate only to
±10 seconds?
This situation is correctly handled using scientific notation. To
report the duration of Jim's walk, we would write it as 3.5 x 102
seconds. The implied precision of the measurement is ±0.1 x
102 seconds = 10 seconds. In scientific notation, we separate out
- the order of magnitude of the value; in this case it is 2 orders
of magnitude, representing 102=100.
- the mantissa of the number; i.e., the factor multiplying the order
of magnitude. In this case, the mantissa is 3.5
The precision of the measurement is then implied by the number of digits in
the mantissa.
Writing numbers in scientific notation also makes calculations easier.
You can even do it "on the back of an envelope". Okay, if you're not good
with arithmetic, understanding scientific notation still allows you to
make a simple check whether the number staring at you from your calculator
makes any sense. As an example, suppose we want to calculate the product 37 x
200?
37 x 200 = (3.7 x 101) x (2.00 x 102)
First, we arrange the factors...
37 x 200 = (3.7 x 2.00) x (101 x 102) = (7.40) x
(101+2) = (7.40) x (103) = 7,400
All measured quantities have units. Be sure to not forget them in your
calculations involving measured quantitites! Units are agreed upon
standards for physical quantities. One problem you'll encounter is that there
is more than one system of units! The two most common are
- English system of units (used exclusively in the United States!)
- length measured in inches, feet (1 ft = 12 in), yards (1 yd = 3
ft) and miles (1 mi = 1,760 yd = 5,280 ft).
- volumes are measured in quarts and gallons (1 gal = 4 quarts = 231
in3).
- temperatures are measured in °F.
- there are other physical quantities we'll encounter later...
- SI system of units also known as the metric system (used by all other
countries in the world, and also used by physicists!)
- length measured in meters.
- volumes measured in liters (1 liter = 1,000 cm3)
- mass (quantity of matter) measured in kilograms.
- temperatures are measured in Kelvin
For the SI system, the order of magnitude is oftentimes written using prefixes
to the units. It is important to realize what is a multiplier and what is a
unit. Don't confuse meter (m) with milli-, where the lower case m always
precedes a unit. Many SI units are the first initial of names and are
capitalized (Joule, Newton, Coulomb) but some are not names and are not
capitalized (meter, second, gram). All prefixes that are smaller than 1 are
lower case:
| centi- | c | 10-2
|
| milli- | m | 10-3
|
| micro- | u | 10-6
|
| nano- | n | 10-9 |
Almost all prefixes that are bigger than 1 are uppercase
(the exception is kilo-, 103 ):
| kilo- | k | 103
|
| mega- | M | 106
|
| giga- | G | 109 |
Below, we give some examples of using scientific notation, computing things from measured quantities,
properly accounting for the units.
Unit conversions
It's often the case that we are given a measurement in English units which we
must then convert to SI, before doing further calculations. For example, Sara
takes a car trip and travels 152 miles in 3.0 hours.
a.) What is the distance she travelled in expressed in SI units?
We need unit conversions between miles and (kilo-)meters. From Rossing Table
A.3 (pg. 659), 1 meter = 3.281 ft. We also know that 1 mile = 5,280 feet. So
we set up the equation below...
Notice how the units for miles and feet algebraicly cancel in the equation.
I've used different symbols to cancel the two units: "\" is used to cancel
"ft" appearing in both numerator and denomintor and "/" is used to cancel
"mi".
b.) What is her average speed in SI units?
We take the distance travelled from a.) and divide by the total time of
the trip to find the average speed. We need the conversion factor
between seconds and hours: 1 hour = 3,600 seconds.
Note that the answer has two significant digits, matching the implied
precision of the time measurement.