VECTOR ADDITION
(Alward/Harlow web File: “vector.doc” 1-25-04)
Exp. #4
Name: _____________________ Partners: _______________________Section No. _______
(Note to instructor: In PHYS 23 when this lab is paired with Experiment 1, time may require that Part IV
be skipped.)
Equipment:
Force table 3 x 50 gram (Beck) weight hangers
1 x 20 g 2 x 50 g
1 x 100 g 1 x 200 g Scissors Ruler
Protractor Washer 3 pulleys
Retaining pin
3 x 50 cm lengths of string
I. PURPOSE
The primary purpose of this exercise is to learn how to add vectors using the graphical method of tail-tohead addition and the more exact method of addition of components. A secondary purpose of this
exercise is to introduce the idea of forces as vectors. In addition to the investigating the basic techniques
of vector addition, the student will verify that the vector sum of all external forces acting on an object is
zero if the object is not accelerating.
Three strings attached to a small ring and passing over pulleys will support three different hanging
masses. The weights of these masses are the forces acting on the ring. Each weight is equal to the mass,
m, multiplied by the acceleration due to gravity, g. In the Standard International system of units weight
(force) is measured in Newtons.
II. PROCEDURE
1. Attach three strings to a central ring, and pass each string over a different pulley positioned at the
perimeter of the circular force table. While adjusting the positions and weights, place a short metal rod
or pin in the center hole to prevent the masses from falling. At the ends of the three strings place weight
hangers and weights. Place the masses shown in the table at the end of the strings. Note: the values
shown include the 50-gram mass of the hanger.
2. Calculate the weight of the each of the masses: the weight, in Newtons (N), is given by m times g,
where m is in kilograms and g = 9.8 m/s2. (Note: kg- m/s2 = N). Places these weights F1, F2, and F3 in
the table on the next page.
3. In this lab, all angles are measured counter-clockwise from the x-axis. Place one of the pulleys at the
zero degrees mark; this will be Pulley 1. Pulley 2 should be somewhere between 80-180 degrees, and
Pulley 3 is somewhere in the third quadrant, between 180 and 270 degrees. Adjust the positions of
Pulleys 2 and 3 until the ring remains motionless at the center of the force table, not contacting the
central pin.
1
m1 =
0.120 kg
F1 =
m2 =
N
1 = 0 degrees
0.200 kg
F2 =
m3 =
N
2 =
F3 =
0.250 kg
N
3 =
F1x =
N
F2x =
N
F3x =
N
F1y =
N
F2y =
N
F3y =
N
Sx =
N
Sy =
S=
N
Average Force =
N
Percent Difference:
N
%
Note: It is very important to reposition the strings grip on the washer each time you move the pulleys:
the strings must not approach the washer tangentially; they must be perpendicular to the washer and
point toward the exact center of the washer, otherwise the angular measurements will be false. Check
the strings often; check them now.
Test equilibrium by removing the pin
and deflecting the ring by one or two
inches in different directions; if the
ring returns each time to the center
position, then equilibrium has been
achieved and the sum of the three
forces is zero. If the ring returns to a
position which is not at the center,
test the pulleys for excessive friction
in the bearings. If necessary, adjust
the positions of Pulleys 2 and 3.
4. On the diagram at the right, sketch
arrows for the forces F1 and F2, and
label the angles 1, 2, 3, of the three
vectors.
2
III. ADDITION OF VECTORS USING THE METHOD OF COMPONENTS
The sum of the three forces is called S:
S = F1 + F2 + F3
(1)
However, since the ring is not accelerating, the net force--the "sum"--is zero. Thus,
F1 + F2 + F3 = 0
(2)
The sum of the x-components of the forces is also zero, and the sum of the y-components is zero:
F1x + F2x + F3x = 0
(3)
F1y + F2y + F3y = 0
(4)
The actual measured sums above won't be zero, however, because of measurement or other experimental
error.
Calculate the components
The x-components F1x, F2x, and F3x are found by multiplying the force magnitudes F1, F2, and F3 by the
corresponding cosines of the angular positions. Do this now and record the x-components in the table.
The y-components F1y, F2y, and F3y are found by multiplying the force magnitudes F1, F2, and F3, by the
corresponding sines of the angular positions. Do this now and record the y-components in the table.
Calculating the sum of the forces
Although the actual total force on the ring is zero because the ring is not accelerating, the calculated
value of the total force may not be zero because of inaccuracies in measurement.
The calculated value of the total force can be found using the Pythagorean Theorem:
S2 = Sx2 + Sy2
(5)
3
This may be re-written as
S = [Sx2 + Sy2]1/2
(6)
Calculate S now by first obtaining
Sx = F1x + F2x + F3x
(7)
and
Sy = F1y + F2y + F3y
(8)
Place these values in the table and then calculate S using Equation 6.
Ideally, S will be zero; it won't actually be zero, but it should be small. How small? By itself, the
smallness of the value S cannot give a measure of how accurately this exercise was performed. For
example, if each force had been several hundred newtons, but the measured sum is only 3 newtons, then
S is small enough in comparison to the typical force that we could claim that S is approximately "zero",
as expected. Thus, we need to compare the S value to a number which is representative of the size of the
individual forces:
Add the three force magnitudes and divide by three to obtain the average force; record this average in the
table. Compare the value S to the average by dividing S by the average, and then multiplying by 100 to
get the percentage error.
4
IV. GRAPHICAL ADDITION OF VECTORS
Use broken lines to draw on the next page the three forces F1, F2, and F3 to scale, tails together, just as
they were on the force table, and pointing in the proper directions. Use the scale 1 cm = 0.20 N. Place
vector F1 along the positive x-axis. Label each arrow (F1, F2, F3) and show the magnitude in newtons,
the angular position in degrees, and the length in cm alongside each arrows.
After the three forces have been drawn, use solid lines to add them tail-to-head following this procedure:
Leave F1 where it is and draw a second vector just like F2 with the tail of this re-positioned F2
connected to the head of F1. Before you do this, it will be necessary for you to determine the angle
between the vectors F1 and F2 (this angle is just 2 -1).
Use a protractor to reproduce this angle when you connect F2 to F1 . Repeat this procedure to connect a
re-positioned F3 to the re-positioned F2.
The sum of the three vectors is the arrow which can now be drawn from the tail of F1 to head of the repositioned F3. If you are very lucky and this exercise has worked perfectly, the length of this arrow will
be negligibly small and it will not be necessary for you to attempt to draw this sum vector. If possible,
draw the sum vector and label it S; if S is drawable, measure its length and convert to units of Newtons
and write the value beside the S vector.
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Tail-to-Head Addition of Force Vectors
The cross "+" below indicates the position of the ring and the point of coincidence of the tails of the
vectors.
+.........................................................................
(draw F1 along here)
6