Equilibrant of a system of forces

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!! !!𝑅!

𝜃! = tan 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 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 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 0°. Fill out the table below.

Force

Mass m

[g]

Mass m [kg]

Magnitude mg [N]

Angle θ

[°]

x-

component

[N]

y-

component

[N]

𝑨

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 𝑨. 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.

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

Mass m

[g]

Mass [kg]

Magnitude mg [N]

Angle θ

[°]

x-

component

[N]

y-

component

[N]

𝑨

100g

.100kg

0.98N

0

0.98

0N

𝑩

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 𝑨 and 𝑩. 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.

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 0°. Fill out the table below.

Force

Mass

m[g]

Mass

m[kg]

Magnitude

mg[N]

Angle

θ[°]

X

Component

[N]

y-

component

[N]

𝑨

25

0.025kg

0.0245N

0

0.245

0

𝑩

50

0.050kg

0.49N

30

0.424

0.25

𝑪

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 𝑨 and 𝑩 and 𝑪. 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.

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