Dimensions
versus Units
dimension -
The type of physical quantity.
-
ex: length, time, mass
units -
The type of system/standard used to make the
measurement.
-
ex: meter, hour, kilogram
Neither is a variable, but, treat them as such
algebraically.
Adding or Subtracting
- Only quantities of
the same dimension can be
added together.
- example: 10 feet
plus 3 inches works.
35 mph plus 7.2 lbs makes no
sense.
-
Units must be the same to add the quantities.
- ex:
10 feet plus 3 inches does not equal 13.
Multiplying or Dividing
- Quantities of
different dimensions will commonly
be multiplied and/or divided.
- ex:
10 feet divided by 5 seconds = 2 ft/s.
22 feet times 3 feet = 66 square feet.
- In
order to cancel units, they must be both
of the same dimension and units.
Base
Units vs. Derived Units
There are seven base units (depending on who you
ask.)
In PH201, we’ll use three base units, those for:
length, time, mass.
Derived units are, yes, derived from base units.
There are many, many, many...
speed:
miles/hour, ft/s, furlong/fortnight, m/s
force:
newtons which = kg m/s/s
Systems
of Measure
SI system (metric system)
mass
– kilogram (kg)
length
– meter (m)
time
– second (s)
British Engineering System (English)
mass
– slug
length
– feet
time
– second
Scalar
versus Vector
scalar -
Any physical quantity that can be fully specified
by a magnitude (size) only.
-
May or may not have some physical dimension.
-
ex: distance, time, speed, area, volume, energy
-
speed = 50.3 m/s
-
Add arithmetically.
vector
- Any physical quantity that must be
defined by
both a magnitude AND a
direction.
- Always has some physical dimension
(units)
- ex: displacement, velocity,
acceleration, force
- velocity = 50.3 m/s at 27o
North of East
- In 1-D, direction can be specified
via a sign (+/-).
- Represented in equations as
boldface or with a
small arrow drawn across the top.
- Represented (graphically) by a
scaled arrow in a
specific orientation.
- Add vectorially.
Vector
Addition
Vector addition can be accomplished graphically
(head-to-tail method) or mathematically (resolution of vectors into components
using trigonometry.) You are responsible for both techniques though the
trigonometric solution is more important for work in physics. Either technique
is appropriate for any and all types of vectors.
Head-to-tail Method
1) Define the coordinate axis. (x/y or NSEW or…)
2) Start at any arbitrary starting point
(usually the origin.)
3) From that starting point, draw the first
vector to scale
and
in its proper orientation.
4) The next vector’s starting point is the end
point (head)
of
the previously drawn vector. Draw this vector to
scale
and in its proper orientation (relative to the axis,
NOT
the previously drawn vector.)
REMEMBER: Negative signs reverse the direction
of the vector, that is a 180o turnabout.
5) Continue until all vectors are drawn.
6) The resultant vector is the vector that
starts at the
original starting point and ends at the head of the last
drawn
vector. Draw it, measure its length and angle.
Vector
Addition (continued)
Resolution of Vectors into Components Method
Any vector can be arbitrarily resolved (divided)
into any number of vectors, the sum of which equals the original.
We are interested only in pairs of mutually
perpendicular components, commonly, x and y components.
1) Define the coordinate axis. (NOT always a
trivial task.)
2) Resolve each vector to be added into its x
and y
components. Keep track of the directions
along the axis
with
positive/negative signs.
A chart here is often helpful for novice vector
adders.
3) Add (arithmetically) all of the x components.
This is the
x
component of the resultant vector.
4) Add (arithmetically) all of the y components.
This is the
y
component of the resultant vector.
5) Find the magnitude of the R.V. via the
Pythagorean
theorem.
6) Find the direction of the R.V. (always an
angle) via the
inverse sine, inverse cosine, or inverse tangent function.
NOTE: For the sum of only two vectors, the law
of cosines
and/or law of sines may replace all of the previous steps.