1
Vector Addition of Forces
Objectives: To use the force table to experimentally determine the force that balances
two or more forces. This result is checked by analytically adding two or more forces
using their horizontal and vertical vector components, and then by graphically adding
the force vectors on the force table.
Theory: If several forces are acting on a point, their resultant 𝑅 is given as
𝑅 = 𝐴 + 𝐵 + 𝐶
Rx = Ax + Bx + Cx
Ry = Ay + By + Cy
R = 𝑅 = 𝑅!! + 𝑅!!
𝜃! = tan!!
𝑅!
𝑅!
Then if the equilibrant 𝐸 is a force that brings the system to equilibrium
E+ 𝑅 = 0, this means
𝐸 = −𝑅 (E = R, θE = θR+180°)
This means Ex = -Rx and Ey = -Ry
Note for today’s lab: read the details, discuss with your group, and follow the
instructions systematically. We have done several of these questions in class so now
work by yourselves. If you want more details, look up your textbook or online.
Method: You will hang some mass on the pulley hangers that are attached by a thread.
This means the weight of that mass is a force vertically down. However, the string is
attached to the central ring of the force table, and this means a tension equal to the
weight of the mass is a force acting on the central ring. This means you can set up one
or more forces acting on the central ring, calculate their resultant force (resultant, 𝑅).
Then you can determine what force (Equilibrant, 𝐸) would balance these forces to bring
the system to equilibrium.
Apparatus:
Force table, 4 pulley clamps, 3 mass hangers, 1 mass set, string (or spool of thread)
Force table: A force table is a simple set up that can be used to observe vector addition
and equilibrium. You can attach a (one or more) pulley at the edge of the table, and
hang a mass on a string that goes through this pulley. Hanging mass means a weight is
2
acting downward and the tension on the hanging string is acting upward. However, on
the top of the table, the string is attached to a central ring. This string applies a
horizontal tension to the ring. The central ring is our object of interest and we will
observe the effect of various forces on this ring. You can change the magnitude of the
force by changing the hanging mass.
The table surface has a protractor so you can set up vectors in specific directions.
You can find more information online on how a force table works.
If a mass “m” is hanging over the pulley, the mass has a force downward (= the weight
of the mass, mg). And the tension on the string is upward. The magnitude of the tension
= mg)
(image credit: CCNY CUNY)
Set up the force table such that 0 of the table protractor is on your right (just like x-axis
on a Cartesian coordinate system. This means 0°, 90°, 180°, and 270° should be along
+x, +y, -x, -y of your coordinate system.
(image credit: CCNY CUNY)
Resultant vs. Equilibrant
Resultant force is the vector sum of the individual forces
1. 1
Vector Addition of Forces
Objectives: To use the forcetable to experimentally
determine the forcethat balances
two or more forces. This result is checked by
analytically adding two or more forces
using their horizontal and vertical vector components,
and then by graphically adding
the forcevectors on the forcetable.
Theory: If several forces are acting on a
point, their resultant � is given as
� = � + � + �
Rx = Ax + Bx + Cx
Ry = Ay + By + Cy
R = � = �!! + �!!
�! = tan!!
�!
�!
Then if the equilibrant � is a forcethat brings
the system to equilibrium
E+ � = 0, this means
2. � = −� (E = R, θE = θR+180°)
This means Ex = -Rx and Ey = -Ry
Note for today’s lab: read the details, discuss with
your group, and follow the
instructions systematically. We have done several of
thesequestions in class so now
work by yourselves. If you want more details, look up
your textbook or online.
Method: You will hang somemass on the pulley hangers
that are attached by a thread.
This means the weight of that mass is a force
vertically down. However, the string is
attached to the central ring of the forcetable,
and this means a tension equal to the
weight of the mass is a forceacting on the
central ring. This means you can set up one
or more forces acting on the central ring,
calculate their resultant force(resultant,�).
Then you can determine what force(Equilibrant, �) would
balance theseforces to bring
the system to equilibrium.
Apparatus:
Force table, 4 pulley clamps, 3 mass
hangers, 1 mass set, string (or spool of
thread)
Force table: A forcetable is a simple set up
that can be used to observe vector addition
and equilibrium. You can attach a (one or more)
pulley at the edge of the table, and
hang a mass on a string that goes through this
pulley. Hanging mass means a weight is
3. 2
acting downwardand the tension on the hanging
string is acting upward. However, on
the top of the table, the string is attached to
a central ring. This string applies a
horizontal tension to the ring. The central ring is
our object of interest and we will
observe the effect of various forces on this
ring. You can change the magnitude of
the
forceby changing the hanging mass.
The table surface has a protractor so you can set up
vectors in specific directions.
You can find more information online on how a
forcetable works.
If a mass “m” is hanging over the pulley, the
mass has a forcedownward(= the weight
of the mass, mg). And the tension on the string
is upward. The magnitude of the tension
= mg)
(image credit: CCNY CUNY)
Set up the forcetable such that 0 of the table
protractor is on your right (just like x-axis
on a Cartesian coordinate system. This means
4. 0°, 90°, 180°, and 270° should be along
+x, +y, -x, -y of your coordinate system.
(image credit: CCNY CUNY)
Resultant vs. Equilibrant
Resultant forceis the vector sum of the individual
forces acting on the ring. The
equilibrant is the forcethat brings the system to
equilibrium.
3
(image credit: CCNY CUNY)
Precaution:
(1) Ensure that the central pin on the forcetable is
always attached in place before
and while you hang any mass unless otherwise
specified. Otherwise the mass
can suddenly drop and hurt someone (and also mess your
experiment).
(2) Measure/note the mass of each hanger before
5. you use it.
(3) The forceneeded to balance the forcetable is
not the resultant forcebut the
equilibrant force, which is negative of the
resultant.
Experimental Procedure I: Use of only one force.
Step 1: Calculation only.Do not hang any mass yet;
you will do that in Step II after you
finish your data table below.
You will hang a mass (an example: 100 g) on a
hanger. The angle should be 0°. Fill out
the table below.
Force Mass m
[g]
Mass
m [kg]
Magnitude
mg [N]
Angle θ
[°]
x-
component
[N]
y-
component
6. [N]
�
Resultant
Then we can writethe resultant and the equilibrant
below
Force Magnitude Angle
Resultant
Equilibrant
4
Step 2: now hang the mass for force�. Then apply
the equilibrant forceas you
determined in your data table above.
To check if the system is actually in
equilibrium, remove the central pin (at the
center of
the ring). If your system is actually in
equilibrium, the ring will stay in place
otherwise
the masses will fall off in the direction on any
net force.
Explain your observations.
Experimental Procedure II: Use of two forces.
Step 1: Calculation only.Do not hang any mass yet;
you will do that in Step II after you
7. finish your data table below.
You will hang two masses (an example: 100 g) on a
hanger. The angle should be 0°. Fill
out the table below.
Force Mass m
[g]
Mass
m [kg]
Magnitude
mg [N]
Angle θ
[°]
x-
component
[N]
y-
component
[N]
�
�
Resultant
Then we can writethe resultant and the equilibrant
below
Force Magnitude Angle
Resultant
Equilibrant
8. Step 2: now hang the masses for forces � and �.
Then apply the equilibrant forceas you
determined in your data table above.
To check if the system is actually in
equilibrium, remove the central pin (at the
center of
the ring). If your system is actually in
equilibrium, the ring will stay in place
otherwise
the masses will fall off in the direction on any
net force.
Explain your observations.
Experimental Procedure III: Use of threeforces.
Step 1: Calculation only.Do not hang any mass yet;
you will do that in Step II after you
finish your data table below.
5
You will hang two masses (an example: 100 g) on a
hanger. The angle should be 0°. Fill
out the table below.
Force Mass m
[g]
Mass
m [kg]
9. Magnitude
mg [N]
Angle θ
[°]
x-
component
[N]
y-
component
[N]
�
�
�
Resultant
Then we can writethe resultant and the equilibrant
below
Force Magnitude Angle
Resultant
Equilibrant
Step 2: now hang the masses for forces � and �
and �. Then apply the equilibrant force
as you determined in your data table above.
To check if the system is actually in
equilibrium, remove the central pin (at the
center of
the ring). If your system is actually in
equilibrium, the ring will stay in place
10. otherwise
the masses will fall off in the direction on any
net force.
Explain your observations.
What to include in your lab report:
(1) Your data tables and observations, comments,
and analysis for threeprocedures
you performed.
(2) Draw a free body diagram for the ring in each
case.
(3) Explain why the forces on the central ring
can be measured using the hanging
masses.
(4) Namesof lab partners and specific contributions
each person made.
Where to submit the lab report: on blackboard
where you downloaded these
instructions from.
Vector Addition of Forces
Objectives: To use the forcetable to experimentally
determine the forcethat balances two or more forces.
This result is checked by analytically adding
11. two or more forces using their horizontal and
vertical vector components, and then by
graphically adding the forcevectors on the
forcetable.
Theory: If several forces are acting on a
point, their resultant � is given as
�=�+�+�
Rx = Ax + Bx + Cx
Ry = Ay + By + Cy
R = �= �!!+�!! !!�!
�! = tan �!
Then if the equilibrant � is a forcethat brings
the system to equilibrium
E+�=0, this means
�=−� (E = R, θE = θR+180°)
This means Ex = -Rx and Ey = -Ry
Note for today’s lab: read the details, discuss with
your group, and follow the instructions
systematically. We have done several of these
questions in class so now work by yourselves. If
you want more details, look up your textbook or
online.
Method: You will hang somemass on the pulley hangers
that are attached by a thread. This means
the weight of that mass is a forcevertically
down. However, the string is attached to
the central ring of the forcetable, and this
means a tension equal to the weight of
12. the mass is a forceacting on the central
ring. This means you can set up one or more
forces acting on the central ring, calculate
their resultant force(resultant,�).
Then you can determine what force(Equilibrant, �) would
balance theseforces to bring the system to
equilibrium.
Apparatus:
Force table, 4 pulley clamps, 3 mass
hangers, 1 mass set, string (or spool of
thread)
Force table: A forcetable is a simple set up
that can be used to observe vector addition
and equilibrium. You can attach a (one or
more) pulley at the edge of the table,
and hang a mass on a string that goes through
this pulley. Hanging mass means a weight is
acting downwardand the tension on the
hanging string is acting upward. However, on
the top of the table, the string is
attached to a central ring. This string applies
a horizontal tension to the ring. The central
ring is our object of interest and we will
observe the effect of various forces on
this ring. You can change the magnitude of
the forceby changing the hanging mass.
The table surface has a protractor so you can set up
vectors in specific directions.
You can find more information online on how a
forcetable works.
13. If a mass “m” is hanging over the pulley, the
mass has a forcedownward(= the weight of
the mass, mg). And the tension on the string
is upward. The magnitude of the tension
)
mg
=
)
(
image credit: CCNY CUNY
Set up the forcetable such that 0 of the table
protractor is on your right (just like x-axis on a
Cartesian coordinate system. This means 0°,
90°, 180°, and 270° should be along +x,
+y, -x, -y of your coordinate system.
(image credit: CCNY CUNY)
14. Resultant vs. Equilibrant
Resultant forceis the vector sum of the individual
forces acting on the ring. The equilibrant is
the forcethat brings the system to equilibrium.
(image credit: CCNY CUNY)
Precaution:
(1) Ensure that the central pin on the forcetable is
always attached in place before and while
you hang any mass unless otherwise specified.
Otherwise the mass can suddenly drop and hurt
someone (and also mess your experiment).
(2) Measure/note the mass of each hanger before
you use it.
(3) The forceneeded to balance the forcetable is
not the resultant forcebut the equilibrant force,
which is negative of the resultant.
Experimental Procedure I: Use of only one force.
Step 1: Calculation only.Do not hang any mass yet;
you will do that in Step II after you finish
your data table below.
You will hang a mass (an example: 100 g) on a
hanger. The angle should be 0°. Fill out
the table below.
Force
15. Massm
[g]
Mass m [kg]
Magnitude mg [N]
Angle θ
[°]
x-
component
[N]
y-
component
[N]
�
200g
0.2kg
1.960N
50
1.260
1.501
Resultant
Then we can writethe resultant and the equilibrant
below
16. Force
Magnitude
Angle
Resultant
1.96N
50
Equilibrant
1.96N
230
Step 2: now hang the mass for force�. Then apply
the equilibrant forceas you determined in
your data table above.
To check if the system is actually in
equilibrium, remove the central pin (at the
center of the ring). If your system is
actually in equilibrium, the ring will stay in
place otherwise the masses will fall off in
the direction on any net force.
Explain your observations.
Experimental Procedure II: Use of two forces.
Step 1: Calculation only.Do not hang any mass yet;
you will do that in Step II after you finish
your data table below.
You will hang two masses (an example: 100 g) on a
hanger. The angle should be 0°. Fill out
the table below.
Force
Massm
[g]
Mass[kg]
17. Magnitude mg [N]
Angle θ
[°]
x-
component
[N]
y-
component
[N]
�
100g
.100kg
0.98N
0
0.98
0N
�
75g
.075kg
0.735N
60
0.37
0.64N
Resultant
1.35N
0.64N
Then we can writethe resultant and the equilibrant
below
Force
Magnitude
Angle
18. Resultant
1.5N
25
Equilibrant
1.5N
205
Step 2: now hang the masses for forces� and �.
Then apply the equilibrant forceas you determined
in your data table above.
To check if the system is actually in
equilibrium, remove the central pin (at the
center of the ring). If your system is
actually in equilibrium, the ring will stay in
place otherwise the masses will fall off in
the direction on any net force.
Explain your observations.
Experimental Procedure III: Use of threeforces.
Step 1: Calculation only.Do not hang any mass yet;
you will do that in Step II after you finish
your data table below.
You will hang two masses (an example: 100 g) on a
hanger. The angle should be 0°. Fill out
the table below.
Force
Mass
m[g]
Mass
m[kg]
20. 1.089
1.40
Then we can writethe resultant and the equilibrant
below
Force
Magnitude
Angle
Resultant
1.77N
52
Equilibrant
1.77N
232
Step2: Now hang the masses for forces� and � and�.
Then apply the equilibrant force as you determined in your data
table above.
To check if the system is actually in
equilibrium, remove the central pin (at the
center of the ring). If your system is
actually in equilibrium, the ring will stay in
place otherwise the masses will fall off in
the direction on any net force.
Explain your observations.
What to include in your lab report:
1) Your data tables and observations, comments, and analysis
for three procedures you performed.
2) Draw a free body diagram for the ring in each case.
3) Explain why the forces on the central ring can be
measured using the hanging masses.
