2.1. General Categories of Physical
Quantities
Scalar quantities are those quantities
which are completely specified by their magnitude,
expressed in
some convenient units.
·
easy to handle since
they can be manipulated by ordinary laws of algebra (Usually can be added,
subtracted, multiplied, divided directly)
·
Examples: length,
mass , area, volume, time, density
Vector quantities are those which
require for complete specification, both magnitude and
direction.
·
Direction is just as
important as the magnitude when specifying quantities
·
It always represented
by an arrow.
·
Examples:
displacement, force, acceleration, velocity, momentum
·
The sum of the
vectors is called the RESULTANT vector.
2.2. Addition of Vectors
Methods in Adding Vectors
A.
Graphical Method – also known as Geometrical
Method and require no computation.
a.
Parallelogram Method
(Tail to Tail Method) – used to add only two vectors
b.
Triangle Method (Head
to Tail Method) – use to add only two vectors
c.
Polygon Method (Head
to Tail Method) – use to add more than 2 vectors
To do this accurately, we need to use a ruler and protractor. The general technique for adding two vectors on graph paper is as follows:
1. Start at the origin and draw the first vector (based
on the numbers given you)
2. At the end of the first vector, make a new origin
3. Draw the second vector starting at the new origin
4. Connect the tail of the first vector to the head of
the second
When
you are finished, the last line you have drawn is the resultant. You can then
measure the length and angle of this vector using a protractor and ruler.
B.
Analytical Method (Component Method/Trigonometric Method)
Trigonometry is also important in physics. When you have a right-angled
triangle, the following relationships are true:
This method
is based on the fact that we can specify a vector by specifying its magnitude
in two perpendicular directions. We take these to be the x and y
directions. We call the length of the vector in the x direction as the x-component
of the vector, and similarly for y. The nice thing about this method is
that once we have the x and y components of the vectors we want
to add, adding them is simple. Let us say that we have two vectors, A
and B, with the components labeled as Ax, Ay,
Bx, and By. It should be obvious which
component each symbol stands for. If our resultant, or sum vector, is called R,
with components Rx and Ry, then we have
But how do we get the components if we are given the
angle and direction? We use trigonometry. Consider the diagram:
We know from
trigonometry that
Ax = A cos q
Ay = A sin q
Ax = A cos q
Ay = A sin q
We now need to be
able to go back from the components to the magnitude and direction. We have, again
from trigonometry,
R2 = Rx2 + Ry2
q = tan-1 (Ry / Rx)
R2 = Rx2 + Ry2
q = tan-1 (Ry / Rx)
The process
for adding two vectors, A and B, is thus:
- Find the components of A and B
using (2a) and (2b)
- Add the components together using (1a) and (1b)
- Find the magnitude and direction of R
using (3a) and (3b)
Note: Calculators are funny things. You should always
make a sketch of your addition to see if your values agree with what your
calculator gives. The magnitudes should be correct, but the angles will often
come out wrong. To adjust the angles, note that
cos q = cos (360° - q )
sin q = sin (180° - q )
tan q = tan (180° + q )
The
summarized steps for component method are as follows:
1. Resolve all the given vectors into their x and y
components.
2. Find the algebraic sum of all the x-components (∑X),
and the algebraic sum of all the y-components (∑Y).
3. Find the magnitude of the resultant using the
Pythagorean theorem.
R = sqrt ((∑X)2 + (∑Y)2)
4. determine the angle of the resultant and the specific
direction using the tangent function.
θ = arctan (∑Y /
∑X)
