Lab Activity:
Addition of Vectors
Phys 221
Names: _____________
Addition of Vectors
Purpose: In this lab, we check a special case of Newton’s laws, namely that the sum of
the forces acting on an object in equilibrium is zero. Since forces are vectors, we will
practice doing vector additions.
First, we measure the forces acting on an object in equilibrium (a ring). Then, we check
that adding the forces as vectors gives zero. We do this in two ways: graphically and
using vector components.
Equipment: protractor, ruler, string, 1 ring, 4 mass hangers, set of slotted masses, force
table with 4 pulleys, large sheet of white paper
What to turn in next week:
1. These lab instructions filled out correctly and completely. No typing necessary but
make it clean and legible. Turn in one copy per group. Every group member should check
the work.
2. Attached graphs showing clearly the graphical vector addition that you have done for
your experiments. These graphs should be to scale, not sketched! Everything should be
labeled, your scale should be given (e.g. 1 inch = 1 N, or whatever is a good scale). The
graphs should fill the page reasonably well, don’t make them too small.
Lab Activity:
Addition of Vectors
Phys 221
Names: _____________
Addition of vectors
1. In a short paragraph, describe the experiment. A drawing and a few sentences are
enough (skip this question and go back to it once you have done question 2).
2. Experiment
Part A: Three forces on a ring in equilibrium
a) Place a pulley at the 20.0 mark on the force table and place a total of 100g on
the end of the string. Be sure to include the 50g of the mass hanger in the
total.
Calculate the magnitude of the force produced by the mass and hanger (in
Newtons. Use F=mg and take g=9.8m/s2). Assume three significant figures
for this and for all other calculations of forces in this lab.
Record the value of this force as F1 in Data Table 1 (attached with this
handout).
b) Place a second pulley at the 90.0-degree mark on the force table and place a
total of 200g on the end of the string. Calculate the force produced and record
this force as F2 in Data Table 1.
Lab Activity:
Addition of Vectors
Phys 221
Names: _____________
c) Determine by trial and error the amount of mass needed and the angle at
which it must be placed in order to place the ring in equilibrium. The ring is
in equilibrium when it is centered on the force table and is not touching the
peg in the middle. Jiggle the ring slightly to make sure that it is truly in
equilibrium. Be sure that all the strings lie along a line, which passes through
the center of the ring.
d) For this third mass, calculate the amount of force produced and from the table
markings determine the angle at which the force acts. Record this force as F3
in Data Table 1.
Part B: Four forces on a ring in equilibrium
a) Place a pulley at 30.0 degrees with 150g on it, one at 100.0 degrees with 200g
on it and one at 145.0 degrees with 100g on it.
b) Calculate the force produced by those masses and record them as F4, F5, and
F6 respectively in Data Table 2.
c) Following the procedure outlined in steps c and d in part A, determine the
equilibrium force and record it as F7 in Data Table 2.
3.
Calculations
Part A: Three forces on a ring in equilibrium
a) As a group on the large sheet of paper, find the resultant of the two applied
forces F1 and F2 by a scaled, graphical construction using the parallelogram
method (place the vectors tail to tail).
How is F3 related to that resultant?
Draw F3 on the graph as well.
Record your results in the “graphical solution” table on the “calculations 1”
page.
Later, outside of class, each individual should construct an 8 ½” x 11” version
of the diagram. You will have to make careful use of your protractor and
ruler!
Lab Activity:
Addition of Vectors
Phys 221
Names: _____________
b) Using trigonometry, calculate the components of F1 and F2 and record them
in the “analytical solution” table (take for x-axis the line going through the 0
mark and for the y-axis the line going through the 90 mark).
Add the components and determine the magnitude of the resultant by the
Pythagorean theorem.
Determine the angle of the resultant from the arc tangent of the components.
Record those results in the table.
What are the magnitude, direction, and x and y components of F3 given your
result for the resultant F1 + F2 above? Write your results in the table.
c) Calculate the percentage error of the magnitude of the experimental value of
F3 with the analytical solution for F3. Also calculate the percentage error of
the magnitude of the graphical solution for F3 compared to the analytical
solution. For each of those comparisons, also calculate the magnitude of the
difference in the angle. Record all values on the “calculations 1” page.
Part B: Four forces on an object in equilibrium
a) Find the resultant of the three applied forces F4, F5, and F6 using a scaled
graphical construction. This time use the polygon method (place the vectors
head to tail). Do it on a large sheet of paper just as in Part A.
Record your results in the “graphical solution” table on the “calculations 2”
page.
Draw also F7 on your graph and record its direction and magnitude in the
table.
Each individual should hand in a 8 ½” x 11” reproduction of the above
construction with their lab report.
b) Using trigonometry, calculate the components of all three forces F4, F5, and
F6 , the components of the resultant, and the magnitude and direction of the
resultant. Record them all in the “analytical solutions” table on the
“calculations 2” page.
Write also the results for F7.
c) Make the same error calculations for this problem as asked for in step 3 above
in the calculations for Part A.
Lab Activity:
Addition of Vectors
Phys 221
Names: _____________
4. Could all four pulleys be placed in the same quadrant or in two adjacent quadrants
and still be in equilibrium? Explain.
5. The forces used in this experiment are the weights of known masses, that is, the
forces exerted on these masses by gravity. Bearing this in mind, explain the
functions of the pulleys.
Lab Activity:
Addition of Vectors
Phys 221
Names: _____________
Data Table 1
Notes:
1. calc. stands for calculated, for example you know the mass is 0.1 kg, therefore the force
associated with that is the weight, or mg, or 0.1kg * 9.8 m/s2=0.98N.
2. exp stands for ‘from experiment’. This value you will get directly from your measurement.
Force
Mass (kg)
F1
0.100 kg
F2
measured resultant
F3, i.e. experiment
0.200 kg
Force (N)
calc.
Direction
20.0
calc.
exp.
90.0
calc.
exp.
Data Table 2
Force
Mass (kg)
F4
0.150 kg
Force (N)
calc.
Direction
30.0
calc.
0.200 kg
F5
100.0
calc.
F6
measured resultant
F7, i.e. experiment
0.100 kg
exp.
145.0
calc.
exp.
Lab Activity:
Addition of Vectors
Phys 221
Names: _____________
Calculations 1
Notes:
1. calc. stands for calculated, for example you know the mass is 0.1 kg, therefore the force
associated with that is the weight, or mg, or 0.1kg * 9.8 m/s2=0.98N.
2. graph stands for ‘from your graph’. This value you will get directly from your graph, not some
calculation! Once you have your graphical result you can calculate F3, which should just be the
opposite of F1+F2.
Graphical Solution
Force
Mass (kg)
F1
0.100 kg
Force (N)
calc.
Direction
20.0
calc.
F2
resultant force
F1 + F2
0.200 kg
90.0
calc.
graph
graph
calc.
calc.
calc.
F3
Analytical Solution
Force
Mass (kg)
F1
0.100 kg
Force (N)
calc.
Direction
y-component
calc.
calc.
calc.
20.0
calc.
F2
resultant
force
F1 + F2
x-component
calc.
0.200 kg
calc.
calc.
90.0
calc.
calc.
calc.
calc.
calc.
calc.
calc.
calc.
F3
Error Calculations
Determine your relative error for the magnitude only. To do this take the calculated magnitude as your true
value and find the ratios
F3,anal  F3,exp
F3,anal
F3,anal  F3, graph
F3,anal
Lab Activity:
Addition of Vectors
Phys 221
Names: _____________
Calculations 2
Graphical Solution
Force
Mass (kg)
F4
0.150 kg
Force (N)
calc.
Direction
30.0
calc.
0.200 kg
F5
100.0
calc.
F6
resultant force
F4 + F5 + F6
0.100 kg
calc.
graph
145.0
graph
calc.
calc.
calc.
F7
Analytical Solution
Force
Mass (kg)
F4
0.150 kg
F5
0.200 kg
Force (N)
calc.
Direction
y-component
calc.
calc.
calc.
calc.
calc.
calc.
calc.
calc.
calc.
30.0
calc.
100.0
calc.
F6
resultant force
F4 + F5 + F6
x-component
calc.
0.100 kg
calc.
calc.
145.0
calc.
calc.
calc.
calc.
F7
Error Calculations
Determine your relative error for the magnitude only. To do this take the calculated magnitude as your true
value and find the ratios
F7,anal  F7,exp
F7, anal
F7,anal  F7, graph
F7,anal
Lab Activity:
Addition of Vectors
Phys 221
Names: _____________