Composition of Concurrent Forces (Force Table)

1 Introduction:If a number of nonparallel forces are acting at the same point on a body, it can be shown that they may be replaced by a singleforce which will produce the same effect on the body. Such a force is called the resultant of the original forces. The processof finding this resultant is called the composition of forces. The single force which will hold a system of concurrent forces(forces acting through a common point) in equilibrium is called the equilibrant of the system. It is equal in magnitude to theresultant, but opposite in direction.

The part of the force effective in some particular direction is called a component of the force. The process of findingcomponents of forces in specified directions is called the resolution of forces. The processes of composition and resolutionmay be performed by either of two methods, graphical or analytical . If f (Figure 1) is the force being considered, theanalytical method of finding the components consists of applying the proper trigonometric relations to the triangles formedby the force f and its components. As shown in Figure 1, the forces fx and fy are the horizontal and vertical components,respectively, of the force f .

Figure 1: Force Components

ff

f

y

x

θ

2 Procedure:1. A portion of your apparatus consists of a small ring attached to four strings and held in place on the force table by the

center post (see Figure 2). Set one pulley at the 0◦ position and suspend a 300-gram weight from the end of the string.Note: The weight of the hanger constitutes a portion of the suspended weight. In like manner, suspend a 400-gramweight at the 90◦ position. If this system is to be held in equilibrium, a third force of the proper amount must be appliedat the required direction. Pull on one of the other strings until you determine the required direction, and then mount athird pulley at this position. Now suspend sufficient weight from this position to center the ring around the post. Thisforce, expressed in grams, is the equilibrant of the other two forces. To check for equilibrium and to minimize the effectsof the friction in the pulleys, raise the ring a short distance above the table and release, noting the new position it takes.Record force and angle.

2. Since the two original forces are at right angles to each other, it is a simple matter to compute the resultant by thePythagorean Theorem. Set the computed value of the resultant on the force table at a position directly opposite to thatof the equilibrant, remove the original two forces, and then check to see if the computed resultant balances the equilibrant.At this point review the definition of a resultant. Record force and angle and also E-R

3. Obtain from the instructor an assignment of a concurrent-force arrangement of three forces. Choose some point on yourgraph paper as the origin and, after laying out the coordinate axes, draw to scale a vector diagram of the force system,labeling the forces f1, f2 and f3 and indicate their directions (see Figure 3).

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Figure 2: Force Table

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Figure 3: Method of Resolving Forces into Components

240

120

45

330g

240g

420g

210g

210g

200g

f

f

f

1

2

3

3

4. Construct the horizontal and vertical components of each force. Measure their magnitudes and record on the diagram asshown in Figure 3. Also record on the data form.

5. Determine the vector sum of the horizontal components Σ fx and the vector sum of the vertical components Σ fy. Considerthese sums as single forces; lay them off on a separate set of axes, as shown in Figure 4. Then construct the resultant Rof these two force vectors, measure its magnitude and the angle θ that it makes with the positive direction of the x-axis,and record.

Figure 4: Relation of the Resultant and the Equilibrant

X

Y

θ

E

RF

Fx

y

6. Now place pulleys on the force table at the positions of the three assigned forces and suspend the proper amount ofmass for each. Then, while pulling on the fourth string in the direction required to produce equilibrium, set the pulleyat the required position and add masses to the hanger until equilibrium is established. Remember to include the mass ofthe hangers in recording the forces. Test the system for equilibrium by the method suggested in Step 1. Also check theuncertainty in the magnitude of the experimental equilibrant by finding how much mass must be added to or removed fromthe mass hanger before a movement of the ring position is observed. Likewise, determine the uncertainty in the angularposition by seeing how many degrees you can move the equilibrant pulley to each side before the ring moves off center.In the space provided for data, record the magnitude and direction of the experimental equilibrant each accompaniedby a ±value for the uncertainty. Now compute and record the percent uncertainty in the experimental value of E. Thencompute and record the percent difference between the values of E determined experimentally and analytically.

7. By applying the proper trigonometric relations to each of the forces in the system assigned to you, determine and recordthe horizontal and vertical components. Next find the sums of the horizontal and vertical components Σ fxand Σ fy re-spectively, and record. Note that the resultant vector is the diagonal of the right triangle and determine its magnitude anddirection analytically and record. How do these values compare with those found graphically? If materially different,check by balancing on the force table.

8. Figure 5 represents a system of four forces in equilibrium such as you had on the force table in step 6. By using the forceswhich you had balanced, with their respective directions, sketch a diagram similar to Figure 5 for your system. Thenconstruct a force polygon such as is represented by Figure 6. being very careful to determine the angle measurementscorrectly. If the vector polygon does not close measure the amount of the discrepancy and record it on the figure.

3 Equations:Percent uncertainty in the magnitude of E (Experimental):

Pu =uncertaintyo f measurement

measurement×100

Percent difference Experimental vs. Analytical:

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Pd =|measured−analytical|

analytical×100

Pythagorean Theorem:

H2= A2

+B2

Trig. Formulas:

X = Rcosθ

Y = Rsinθ

R2= A2

+2ABcosθ+B2

θ = arctanYX

Table 1: Composition and Resolution of Assigned ForcesAssigned Magnitude Direction Graphical Analytical

Force method methodHorizontal Vertical Horizontal Vertical

f1

f2

f3

ΣFx;ΣFy

Resultant,R

Equilibrant,E

Figure 5: A System of Forces in Equilibrium

f

f

f

f

1

2

3

4

5

Figure 6: Force Polygon

f

f

f

f

1

2

3

4

4 Questions:1. State what part of this experiment best verified the definition of a resultant of a system of forces. Explain just how the

definition was verified.

2. State clearly the relationship between the resultant and equilibrant of a system of forces.

3. Can the effectiveness of any single force be truly represented by two components of the force? Qualify your answer bythe results of this experiment.

4. If the masses of the mass hangers are exactly the same, could their masses have been neglected? Explain.

5. Was the ring in equilibrium when it was not centered around the post on the force table? Explain why it was necessaryfor it to be centered in this experiment.

6. If the ring had weighed considerably more, what would have been the resulting effect on the system?

7. By what amount, if any, did your polygon fail to close? Can this amount of discrepancy be justified by the uncertainty inthe experimental value of the equilibrant? Explain.

8. To which did the equilibrium of the system seem to be more sensitive, small changes in force or in angle? From anexamination of the setup, what explanation can you give for this?

9. Two forces of 200 and 300 grams, respectively, make an angle θ of 50◦ with each other. Find the resultant analyticallyby applying the cosine law, namely, R2

= A2+2ABcosθ+B2 where A and B are the two forces and R is the resultant.

10. If the two forces in Question 9 make an angle of 90◦ with each other, find the resultant. To what form does the cosinelaw reduce to in this case?

11. If the two forces in Question 9 make an angle of 0◦ with each other, find the resultant. Reduce the cosine law equation tothe simplest form.

12. Does the percent difference between the experimental and analytical values of E seem to be justifiable for the experimentalsetup you used? Explain your reasoning.

13. In this experiment you used mass units on force quantities. In most cases this would be considered incorrect and wouldlead to incorrect results when used in computations. Why were you able to do this in this experiment? Show a samplecalculation that indicates the use of correct units and demonstrate why this was not necessary for these computations.

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