EXPERIMENT 2
SUMMING VECTOR QUANTITIES USING PARALELLOGRAM METHOD
Purpose : Summing the vector quantities using the parallelogram method
Apparatus:
Different masses between 1-1000 grams
A flat wood,
Two pulleys,
A few millimetric sheets.
Prior information:
Some quantities can be expressed in numbers, but there are some quantities that numbers are
not sufficient to express them. In some cases, addition to numbers, the directions of the
quantities should be given. Therefore, physical quantities are divided in two: scalar and vector
quantities.
Scalar Quantities:
There is no direction in the question for physical quantities such as mass, energy, temperature,
work, electric charge, time, volume…. We would have enough information when their
numerical values and units are given. These quantities are called scalar quantities.
Vector Quantities:
Some quantities such as velocity, force, acceleration, displacement are directional quantities.
The quantities of this type cannot be expressed by just their numerical values and units. The
quantities that are expressed by their magnitudes, starting points and directions are called
vector quantities.
When we say “a train goes with a speed of 30 km/hour”, it is said that the event is not
expressed clearly. A question then arises: to which direction? For example, if we say “a train
goes with a speed of 30 km/hour to north” the event would be expressed clearly.
Vector representation:
Ending
point
Starting
point
0
‫ܨ‬Ԧ 1
Magnitude
A
As can be seen from the figure above, vector quantities are represented by a directional
line segment. This vector has three elements:
Application (starting) point: This is the point where the vector is applied or this point is the
starting point. The application point of the above vector is at 0.
ሬԦ
‫ܭ‬
Magnitude: The magnitude of a vector is the numerical
value of it. As the figure on the right shows the magnitude of
the K vector is four units.
ሬԦ
‫ܭ‬
0
west
Direction: The direction of the arrow on the edge of a vector is defined as vector’s direction.
In the figure, the direction of the K vector is from 0 to A or to east.
Equivalence of Two Vectors: Two vectors with the same
directions and magnitudes are said to be equal to each other.
In the figure, K and L vectors are equal, since their
directions (east) and magnitudes (three units) are equal
(K=L).
ሬԦ
‫ܭ‬
Negative of a vector: A vector which has the same
magnitude, but in the opposite direction of the K vector, is
said to be a negative of the vector.
ሬԦ
‫ܭ‬
Vector Shift: It is possible to shift a vector quantity without
changing its magnitude and direction. If a vector is shifted
by changing its direction, then it is a different vector.
‫ܮ‬ሬԦ
ሬԦ
െ‫ܭ‬
ሬԦ
‫ܭ‬
ሬԦ
‫ܭ‬
A
east
Vector Sum: There are several methods used to sum vectors: endwise addition (polygon)
method and parallelogram method.
Polygon Method: According to this method, the when two vectors are summed without
changing their magnitude and direction, the ending point of one vector is joined by the
starting point of the other vector. After that, a new vector whose starting and ending points
are drown from the starting point of the first vector to the ending point of the second vector.
This new vector is just the sum of two vectors.
ሬԦ
‫ܭ‬
‫ܮ‬ሬԦ
‫ܮ‬ሬԦ
Figure I.
ሬԦ ൅ ‫ܮ‬ሬԦ
‫ܭ‬
ሬԦ
‫ܭ‬
Figure II.
Parallelogram method: In this method (follow Fig. III and Fig. IV), the starting points of
two-vectors are joined first. A line starting from the ending point of K vector is drawn as to be
parallel to vector L and another line starting from the ending point of L vector is drawn as to
be parallel to vector K. A new line starting from the joined starting points of K and L vectors
to the joined edges of two parallel lines is drawn. This new vector is again the sum of K and L
vectors.
ሬԦ
‫ܭ‬
‫ܮ‬ሬԦ
Figure III.
ሬԦ
‫ܭ‬
ሬԦ ൅ ‫ܮ‬ሬԦ
‫ܭ‬
‫ܮ‬ሬԦ
Figure IV.
Note: When two or more vectors are summed, all the vectors involved must have the same
unit. For example, two sum a velocity vector and a displacement vector is meaningless.
EXPERIMENT:
Build an experimental set-up as shown in figure:
Q
P
E
An experimental set-up composed of two pulleys and three masses.
Attach the millimetric sheet on the flat wood as to be behind the set-up.
Connect a rope properly and pass it through the pulleys.
Then, attach three masses (P, E and Q) to the free edges of the rope and release the system.
Mark the rope path on sheet and measure the direction and angles in between.
For the stationary case, find the composite force and put it on the table below. The force
should be in N (Newton) units and compare this force with that E mass.
Find the composite force of the stationary system using the cosines theorem and put it on
table.
Using different masses and sheets repeat this experiment twice.
Exp. No
1
2
3
E (N)
Q (N)
P (N)
Angle
(Degrees)
Rn (N)
Rn (N)
Experimental Theoretical
%
difference
ERROR ANALYSIS:
Perform an error analysis using the exact values and the results found from experiment. Using
the cosines theorem, evaluate the relative error formula for the composite force of the
stationary system and calculate the relative error for one measurement. Write down the
sources of errors.
RESULTS and COMMENTS:
Write down the results and comments about this experiment.
REFERENCES:
Fundamentals of Physics, D. Halliday and R. Resnick, Page: 247-249.
REPORT
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