Composition Of Concurrent Forces[1]force Table

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 single force which will produce the same effect on the body. Such a force is called the resultant of the original forces. The process of 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 the resultant, but opposite in direction. The part of the force effective in some particular direction is called a component of the force. The process of finding components of forces in specified directions is called the resolution of forces. The processes of composition and resolution may be performed by either of two methods, graphical or analytical . If f (Figure 1) is the force being considered, the analytical method of finding the components consists of applying the proper trigonometric relations to the triangles formed by the force f and its components. As shown in Figure 1, the forces f x and fy are the horizontal and vertical components, respectively, of the force f . Figure 1: Force Components

f fy θ fx

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-gram weight at the 90◦ position. If this system is to be held in equilibrium, a third force of the proper amount must be applied at the required direction. Pull on one of the other strings until you determine the required direction, and then mount a third pulley at this position. Now suspend sufficient weight from this position to center the ring around the post. This force, expressed in grams, is the equilibrant of the other two forces. To check for equilibrium and to minimize the effects of 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 the Pythagorean Theorem. Set the computed value of the resultant on the force table at a position directly opposite to that of 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 your graph paper as the origin and, after laying out the coordinate axes, draw to scale a vector diagram of the force system, labeling the forces f 1 , 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

f

2

330g

210g

f1 120 45 240

200g

210g

240g

f3

420g

3

4. Construct the horizontal and vertical components of each force. Measure their magnitudes and record on the diagram as shown in Figure 3. Also record on the data form. 5. Determine the vector sum of the horizontal components Σ f x and the vector sum of the vertical components Σ f y . Consider these sums as single forces; lay them off on a separate set of axes, as shown in Figure 4. Then construct the resultant R of 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

Y Fx Fy

R

θ X E

6. Now place pulleys on the force table at the positions of the three assigned forces and suspend the proper amount of mass for each. Then, while pulling on the fourth string in the direction required to produce equilibrium, set the pulley at the required position and add masses to the hanger until equilibrium is established. Remember to include the mass of the hangers in recording the forces. Test the system for equilibrium by the method suggested in Step 1. Also check the uncertainty in the magnitude of the experimental equilibrant by finding how much mass must be added to or removed from the mass hanger before a movement of the ring position is observed. Likewise, determine the uncertainty in the angular position 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 accompanied by a ±value for the uncertainty. Now compute and record the percent uncertainty in the experimental value of E. Then compute 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 record the horizontal and vertical components. Next find the sums of the horizontal and vertical components Σ f x and Σ fy respectively, and record. Note that the resultant vector is the diagonal of the right triangle and determine its magnitude and direction 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 forces which you had balanced, with their respective directions, sketch a diagram similar to Figure 5 for your system. Then construct a force polygon such as is represented by Figure 6. being very careful to determine the angle measurements correctly. 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 =

uncertainty o f measurement × 100 measurement

Percent difference Experimental vs. Analytical: 4

Pd =

|measured − analytical| × 100 analytical

Pythagorean Theorem: H 2 = A2 + B2 Trig. Formulas: X

= R cos θ

Y

= R sin θ

R2 = A2 + 2AB cosθ + B2 θ = arctan

Assigned Force

Magnitude

Y X

Table 1: Composition and Resolution of Assigned Forces Direction Graphical Analytical method method Horizontal Vertical Horizontal Vertical

f1 f2 f3 ΣFx ; ΣFy Resultant, R Equilibrant, E

Figure 5: A System of Forces in Equilibrium

f2 f1 f4 f3

5

Figure 6: Force Polygon

f2 f3 f1 f4

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 by the 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 necessary for 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 in the 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 an examination 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 analytically by applying the cosine law, namely, R2 = A2 + 2AB cosθ + 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 cosine law 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 to the simplest form. 12. Does the percent difference between the experimental and analytical values of E seem to be justifiable for the experimental setup 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 would lead to incorrect results when used in computations. Why were you able to do this in this experiment? Show a sample calculation that indicates the use of correct units and demonstrate why this was not necessary for these computations.

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