Vector addition purpose

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  1. Purpose

The purpose of this experiment is to use the force table to experimentally determine the force which balances two other forces. This result is checked by adding the two forces using their components and by graphically adding the forces.

  1. Theory

Many physical quantities can be completely specified by their magnitude alone. Such quantities are called scalars. Examples include such diverse things as distance, time, speed, mass and temperature. Another physically important class of quantities is that of vectors, which have direction as well as magnitude.

A-) Experimental Method: Two forces are applied on the force table by using masses over pulleys positioned at certain angles. Then the angle and mass hung over a third pulley are adjusted until it balances the other two forces. This third force is called the equilibrant () since it is the force which established equilibrium. The equilibrant is not the same as the resultant (). The resultant is the addition of the two forces. While the equilibrant is equal in magnitude to the resultant, it is in the opposite direction because it balances the resultant (see Fig.1.1). So the equilibrant is the negative of the resultant:




- (1.1)

Figure 1.1: The equilibrant balances the resultant

B-) Component Method: Two forces are added together by adding the x- and y-components of the forces. First the two forces are broken into their x- and y-components using trigonometry:

and (1.2)












where Ax is the component of the vector and is the unit vector in the x-direction as shown Fig. 1.2. To determine the sum of and , the components are added to get the components of the resultant .

Figure 1.2: Vector Components
To complete the analysis, the resultant force must be in the form of a magnitude and a direction (angle). So the components of the resultant (Rx and Ry) must be combined using the Pythagorean theorem since the components are at right angles to each other:
and using trigonometry gives the angle:


C-) Graphical Method: Two forces are added together by drawing them to scale using a ruler and protractor. The second () is drawn with its tail to the head of the first force (). The resultant () is drawn from the tail of the to the head of as shown in Fig.1.3. Then the magnitude of the resultant can be measured directly from the diagram and converted to the proper force using the chosen scale. The angle can also be measured using the protractor.

Figure 1.3: Adding vectors head to tail


  1. Assemble the force table as shown in the Assemble section. Use three pulleys (two for the forces that will be added and one for the force that balances the sum of the two forces).

  2. If you are using the Ring Method, screw the center post up so that it will hold the ring in place when the masses are suspended from the two pulleys. If you are using the Anchor String Method, leave the center post so that it is flush with the top surface of the force table. Make sure the anchor string is tied to one of the legs of the force table so the anchor string will hold the strings that are attached to the masses that will be suspended from the two pulleys.

  3. Hang the following masses on two masses on two of the pulleys and clamp the pulleys at the given angles:

Force = 500 N at 0o (1.6)

Force = 1000 N at 120o (1.7)

Experimental Method:

By trial and error, find the angle for the third pulley and the mass which must be suspended from it that will balance the forces exerted on the strings by the other two masses. The third force is called the equilibrant since it is the force which establishes equilibrium. The equilibrant is the negative of the resultant:

- (1.8)

Record the mass and angle required for the third pulley to put the system into equilibrium in Table 1.1. (page.16)

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