Phys 221
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
To Do: Follow the worksheet
Your write-up is due on the due date indicated on class calendar on the web site (no
late write-up will be accepted).
Phys 221
Addition of vectors
1. 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=9.80 m/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.
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.
Phys 221
2. 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, construct an 8 ½” x 11” version of the diagram. You
will have to make careful use of your protractor and ruler!
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.
Phys 221
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.
Include a 8 ½” x 11” reproduction of the above construction in your 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.
Phys 221
Data Table 1
Mass (kg)
Force

F1

F2
Force (N)
Direction
0.100 kg
20.0
0.200 kg
90.0
measured resultant


 
F3   F1  F2

Data Table 2
Mass (kg)
Force

F4

F5

F6
measured resultant


  
F7   F4  F5  F6

Force (N)
Direction
0.150 kg
30.0
0.200 kg
100.0
0.100 kg
145.0
Phys 221
Calculations 1
Graphical Solution
Mass (kg)
Force

F1

F2
Force (N)
Direction
0.100 kg
20.0
0.200 kg
90.0
sum of the forces
F  F 


1
2


 
F3   F1  F2

Analytical Solution
Mass (kg)
Force

F1

F2
20.0
0.200 kg
90.0
F  F 

1
2


 
F3   F1  F2
Direction
0.100 kg
sum of the forces

Force (N)

Error Calculations
x-component
y-component
Phys 221
Calculations 2
Graphical Solution
Mass (kg)
Force

F4

F5

F6
Force (N)
Direction
0.150 kg
30.0
0.200 kg
100.0
0.100 kg
145.0
sum of the forces


  
F4  F5  F6

  
F7   F4  F5  F6


Analytical Solution
Mass (kg)
Force

F4

F5

F6



Direction
0.150 kg
30.0
0.200 kg
100.0
0.100 kg
145.0
sum of the forces
  
F4  F5  F6

  
F7   F4  F5  F6
Force (N)

Error Calculations
x-component
y-component
Phys 221
Write-up
1. In a short paragraph, describe the experiment. A drawing and a few sentences are enough
2. Three forces on a ring.
2.1. Data table: explain what you measured.
2.2. Tail to tail vector diagram and graphical table: explain how you drew the vectors and
what you measured on your vector diagram.
2.3. Analytical table: explain how you did your computations.
2.4. Error calculations
3. Four forces on a ring.
3.1. Data table: explain what you measured.
3.2. Head to tail vector diagram and graphical table: explain how you drew the vectors and
what you measured on your vector diagram.
3.3. Analytical table: explain how you did your computations.
3.4. Error calculations
4. Answer these two questions
4.1. Could all four pulleys be placed in the same quadrant or in two adjacent quadrants and
still be in equilibrium? Explain.
4.2. 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.
5. Turn in one typed report per group on the due date (listed on the class web site).