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…

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 π is given as
π=π΄+π΅+πΆ
Rx = Ax + Bx + Cx
Ry = Ay + By + Cy
R = π= π!!+π!! !!π!
π! = tan π!
Then if the equilibrant πΈ is a force that brings the system to equilibrium
E+π=0, this means
πΈ=βπ (E = R, ΞΈE = ΞΈR+180Β°)
This means Ex = -Rx and Ey = -Ry
Note for todayβ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, π).
Then you can determine what force (Equilibrant, πΈ) 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 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 βmβ 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 0Β°, 90Β°, 180Β°, and 270Β° 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 acting on the ring. The equilibrant is the force that brings the system to equilibrium.
(image credit: CCNY CUNY)
Precaution:
(1) Ensure that the central pin on the force table 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 force needed to balance the force table is not the resultant force but 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 0Β°. Fill out the table below.
Force
Mass m
[g]
Mass m [kg]
Magnitude mg [N]
Angle ΞΈ
[Β°]
x-
component
[N]
y-
component
[N]
π¨
200g
0.2kg
1.960N
50
1.260
1.501
Resultant
Then we can write the resultant and the equilibrant below
Force
Magnitude
Angle
Resultant
1.96N
50
Equilibrant
1.96N
230
Step 2: now hang the mass for force π¨. 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.
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 0Β°. Fill out the table below.
Force
Mass m
[g]
Mass [kg]
Magnitude mg [N]
Angle ΞΈ
[Β°]
x-
component
[N]
y-
component
[N]
π¨
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 write the resultant and the equilibrant below
Force
Magnitude
Angle
Resultant
1.5N
25
Equilibrant
1.5N
205
Step 2: now hang the masses for forces π¨ and π©. 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.
Experimental Procedure III: Use of three 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 0Β°. Fill out the table below.
Force
Mass
m[g]
Mass
m[kg]
Magnitude
mg[N]
Angle
ΞΈ[Β°]
X
Component
[N]
y-
component
[N]
π¨
25
0.025kg
0.0245N
0
0.245
0
50
0.050kg
0.49N
30
0.424
0.25
πͺ
125
0.125kg
0.1225N
70
0.42
1.15
Resultant
1.089
1.40
Then we can write the resultant and the equilibrant below
Force
Magnitude
Angle
Resultant
1.77N
52
Equilibrant
1.77N
232
Step2: Now hang the masses for forces π¨ and π© and πͺ. 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.