PHYSICS 1421  SECTION:                                       

 

 

NAME:                                                           ID#:                                        

 

 

 

Vectors-Force Table (Vector Addition)

 

 

 

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 by using their components and by graphically adding the forces.

 

Theory

 

This  experiment  finds  the  resultant  of  adding  two  vectors  by  three  methods:

 

experimentally, by components, and graphically.

 

➤ NOTE: In all cases, the force caused by the mass hanging over the pulley is found by multiplying the mass by the acceleration due to gravity.

 

 

Equipment

 

 

 

 

 

 

 

 

 

 

 

Figure 1 Force table

 

Figure 2 Mass Hangers and Spool of Thread

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3 Pulleys and Plastic Ring

 

 

 

 

Experimental Method

 

Two forces are applied on the force table by hanging 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 (FE) since it is the force which establishes equilibrium. The equilibrant is not the same as the resultant (FR). 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 Figure 4). So the equilibrant is the negative of the resultant:

 

 

 

 

 

 

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         

 

where

Ax   is the x-component of vector FA  and x is the unit vector in the x-direction. See

Figure  5.  To  determine  the  sum  of  FA   and  FB,  the  components  are  added  to  get  the components of the resultant FR :

 

 

 

 

 

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:

 

 

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Â

 

FR    =                                           and the angle given by         Ɵ   =   tan-1  

 

 

 

 

Graphical Method

 

 

Two forces are added together by drawing them to scale using a ruler and protractor. The second force (FB ) is drawn with its tail to the head of the first force (FA). The resultant (FR) is drawn from the tail of FA to the head of FB. See Figure 6. 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.

 

 

 

 

 

 

 

Setup

 

 

1.   Assemble  the  force  table  as  shown  in  the  Assembly  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 of the pulleys and clamp the pulleys at

 

the given angles:

 

           

Procedure (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 (FE ) since it is the force which establishes equilibrium. The equilibrant is the negative of the resultant:

 

Record the mass and angle required for the third pulley to put the system into equilibrium in Table 1.

 

To determine whether the system is in equilibrium, use the following criteria.

 

 

 

 

Ring Method of Finding Equilibrium

 

 

The ring should be centered over the post when the system is in equilibrium. Screw the center post down so that it is flush with the top surface of the force table and no longer able to hold the ring in position. Pull the ring slightly to one side and let it go. Check to see that the ring returns to the center. If not, adjust the mass and/or angle of the pulley until the ring always returns to the center when pulled slightly to one side.

 

Anchor String Method of Finding Equilibrium

 

 

The knot should be centered over the hole in the middle of the center post when the system is in equilibrium. The anchor string should be slack. Adjust the pulleys downward until the strings are close to the top surface of the force table. Pull the knot slightly to one side and let it go. Check to see that the knot returns to the center. If not, adjust the mass and/or angle of the third pulley until the knot always returns to the center when pulled slightly to one side.

 

Analysis

 

To determine theoretically what mass should be suspended from the third pulley, and at what angle, calculate the magnitude and direction of the equilibrant (FE) by the component method and the graphical method.

 

 

 

Analytical / Component Method

 

 

Add the vector components of Force A and Force B to determine the magnitude of the equilibrant. Use trigonometry to find the direction (remember, the equilibrant is exactly opposite in direction to the resultant). Record the results in Table 1.

        Force A = 75g at 40^o

        Force B = 150g at 120^o