Physics Notes
Variables, Quantities and Units
Parts of a Measurement: Let’s examine a
common type of measurement. Consider the
width of this paper. If you measure it with a
ruler you would see it is eight and a half
inches wide. This can be expressed as:
Conversions: Everything would be great if
there was just one unit per type of
measurement, but unfortunately that’s not
the way it is. Each type of quantity can be
measured in many different units.
For
example, distance can be measured in inches,
feet, yards, miles, meters, kilometers,
furlongs, fathoms, links, rods, chains, leagues,
cables, nautical miles, etc. Often we mix units.
For example you may express your height as
70 inches or as 5 feet, 10 inches.
w = 8.5 Inches
That statement says three things about your
measurement.
1. The variable w represents the quantity
width.
2. 8.5 is the quantity of the measurement.
3. The unit of measure is inches.
In this country, we are more familiar our
customary system of measurement. But in
science we use the SI system that almost all
the rest of the world uses. The meter is the
basic unit of length. The basic unit of time is
the second, and the basic unit of mass is the
kilogram. You will become familiar with
more units for different types of quantities as
we progress through physics.
In each of the next two paragraphs, two
measurements are mentioned. In the space
below each, write what quantity is present, its
value and its units. The first is done as an
example:
Andrea drives from home to school a
distance of 10 miles. It takes her 15 minutes
to get there.
distance
________________
time
________________
10
Quantity: ________________
15
________________
miles
________________
minutes
________________
Variable:
Unit:
One nice thing about the metric system is that
most units are in factors of ten, so that
converting between quantities of expressing
in scientific notation is relatively simple.
Some problem we have may requires us to
convert between types of units. Shown below
is a partial table of conversion factors:
Jesse drops a coin from the 2nd floor of a
building 22 feet above the ground. It takes
1.2 seconds for it to hit the ground.
(distance)
Variable: height
________________
time
________________
22
Quantity: ________________
1.2
________________
feet
________________
seconds
________________
Unit:
100 cm = 1 meter
1000 mm = 1 meter
1 km = 1000 meters
12 inches = 1 foot
3 feet = 1 yard
5280 feet = 1 mile
2.54 cm = 1.00 inch
UNITS ARE IMPORTANT!!!
For example, how old are you? _______________
1
60 sec = 1 min
60 min = 1 hour
24 hours = 1 day
365.25 days = 1 year
1.00 pound = 4.45 N
The distances (left column) are in 3 groups
Now you work the other two conversions:
914.4 cm
10 yards = ________
SI units, American units, both systems
Why? _________________________________________
The only distance you see with a decimal
point is 2.54 cm = 1.00 inch.
10 yd _3 ft_ 12 in 2.54 cm
1
1 yd 1 ft
1 in
integer if converting between systems
Why? Not
_________________________________________
The Factor-Label Method:
following examples:
Consider the
24.14 km
15 miles = ________
100 inches = ________ meters
15 mi 5280 ft 12 in 2.54 cm 1 m__ 1 km__
1
1 mi
1 ft
1 in 100 cm 1000 m
10 yards = ________ cm
15 miles = ________ km
Measurements: 6.0 cm is not the same as
6.00 cm. To understand what I mean,
consider this example.
Start with the quantity you want to convert
on the left and express it as a ratio. (The
denominator will usually be a unitless “1”
unless you are converting a mixed unit):
Three students were tasked to measure
various objects. Student A used a standard 30
cm ruler that had ten little marks between
each cm. Student B used a 30 cm ruler that
didn’t have the mm marks on it. Student C
didn’t have a ruler, so he used a piece of
paper that had a mark 10 cm from one edge.
The table below shows the results of their
measurements:
100 inches
1
Look at the unit you want to get rid of in the
first cell and place that in the bottom of the
second term:
100 inches 100 inches
1
inches
Try to find a conversion factor that will get
you to the desired units. If it can’t be done in
one step, map out a path you the unit you
want. In this case:
Items
Width of paper
Height of book
Radius of CD
100 inches 2.54 cm
1
1.00 inches
Cancel out units (one on top, one on bottom):
B
21.6 cm
5.6 cm
6.0 cm
C
22 cm
6 cm
6 cm
All measurements carry with them
information about what kind of instrument
was used to make them. Note that we can
usually estimate the distance between the
smallest markings of what you’re using. The
first student was able to measure down to a
tenth of a millimeter, the second to the
millimeter, and the best the third student
could do was the nearest cm.
Any
calculations done using these measurements
can be no more accurate than the initial
measurements themselves.
100 inches 2.54 cm
1
1.00 inches
Repeat until you have the units you are
converting to:
100 inches 2.54 cm
1m
1
1.00 inches 100 cm
Multiply the numerators and divide by the
denominators and you have your answer:
100 x 2.54 x 1 =
1.00 x 100
A
21.58 cm
5.65 cm
6.00 cm
2.54 m
2
KEY
Name: ___________________________
Homework
Variables, Quantities and Units
1. Name all variables, quantities and units in the following examples:
a. During a lab, a ball was dropped 200 cm and took .64 seconds to hit the floor.
height
____________________
time
____________________
200
Quantity: ____________________
.64
____________________
Unit:
seconds
____________________
Variable:
centimeters
____________________
b. An electric car starts at rest and accelerates 2.0 m/s2 until it reaches a speed of 10 m/s. It
then travels 50 m at a constant velocity.
acceleration
____________________
speed
____________________
distance
____________________
2.0
Quantity: ____________________
10
____________________
50
____________________
Unit:
m/s
____________________
meters
____________________
Variable:
m/s2
____________________
2. Convert the following quantities:
a. 12.0 inches = ______ cm
12.0 in
1
2.54 cm
= 30.5 cm
1 in
b. 1 year = ________ seconds
1 yr 365.25 day 24 hr 60 min 60 sec
1
1 yr
1 day 1 hr
1 min = 31,557,600 sec
30.48 feet (1 nm = 1852 meters)
c. 1 nautical mile = ________
1 nm 1852 m 100 cm __1 in__ _1 ft_
= 6076.12 ft
1
1 nm
1m
2.54 cm 12 in
3. A student is trying to calculate the acceleration of a dragster on a
quarter-mile track. He times three trials using his watch and gets 5.1,
5.2 and 4.9 seconds respectively. He calculates the average as
5.06667 seconds. Is anything wrong with his answer, and if so, what?
If the measured times are only precise to one tenth of a
second, how can the average be precise to one hundredthousandth of a second? This question is to get you to
start thinking about significant figures!
NOTE: ‘b’ and ‘c’
answers are incorrect
for significant figures,
but we’ll get to that
later!