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Application of Graphical and Analytical Vector Addition to Forces at Equilibrium Agsaoay , M. L., Belino , S., Galas , A., Iddoba , F., & Magbalita , M.*
Department of physical sciences College of Science University of the Philippines Baguio Governor Pack Road, 2600 Baguio City *Corresponding author:
[email protected]
Abstract A vector quantity, such as force, consists of both magnitude and direction. In determining the resultant force of two or more forces, graphical and analytical vector addition may be applied. In the experiment, a method referred to as “Keep The Ring At The Center” (KTRATC) was used to observe forces at equilibrium, where the ring is maintained at the center by a net force of zero using pulleys and hanging masses attached to the ring through strings. A single vector or mass was considered an equilibrant which has the same magnitude but the opposite direction of the resultant, or the sum, of all the other vectors, thus maintaining balance in the system. The magnitude and direction of the equilibrant can thus be derived from the resultant vector. This was done in the experiment to verify the experimental value of the equilibrant and to calculate the mass of an unknown block of wood. It was observed in the results of the experiment that the analytical and graphical technique each has its accuracy in different aspects. Keywords: vector, resultant, equilibrant
Results and Discussions In the experiment, hanging masses were connected to a ring on the force table using pulleys and strings of equal lengths. The addition of vectors was performed analytically and graphically using data gathered in the method “Keep The Ring At The Center” (KTRATC) where the goal was to keep the ring at the center of the force table by changing the hanging masses, or the magnitudes, and adjusting the angles, or the directions. Keeping it at the center signifies static equilibrium within the system, which means that the sum of all forces acting on the ring is zero, resulting to no acceleration [1]. It is also important not to disturb the hanging masses while conducting the KTRATC because doing so affects the equilibrium that was already achieved by the system since other external forces are being added to it. For instance, slightly pushing one of the hanging masses upward results to movement of the ring in the direction of the other hanging masses due to the additional pushing force and normal force on the hanging mass that affects the value of the tension [2] Moreover, due to the friction present between the pulleys and the strings, the disturbed system may not be able to restore balance easily. Aside from the limitations due to resources, there is no limit in the number of hanging masses that may be used to KTRATC. As long as the net force of the hanging masses added do not exceed that of the equilibrant, or the hanging mass 1, any number of other forces may be used. However, maximizing the number of hanging masses used may also increase the errors in the data because of the friction of the pulley and the string which is disregarded in the calculations. Along the x-axis, the net force on the ring consists of the tension force(s) exerted by the string(s) attached. For each hanging mass, the net force consists of the gravitational force, given by equation 2.1 where m is the mass and g is acceleration due to gravity 9.80 m/s 2 downwards, and the tension force exerted by the string. The string connecting the ring and the hanging mass has the same tension force, thus the tension force acting on the ring can
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also be given by equation 2.1 [2]. However, since all strings attached to the rings would then have a tension incorporating the value of g , for experimental purposes, the magnitude considered was only the mass. Therefore, the principles of vector addition are still applicable.
(Eq. 2.1)
The equilibrant is a single force that must be combined with a set of forces to create a system of static equilibrium, where all forces are balanced, resulting to zero net force, thus no movement or acceleration. In the experiment, the hanging mass 1 is considered the equilibrant since in all cases, the hanging masses or forces added on the force table are balanced by the hanging mass 1, by exerting on the ring an equal magnitude of force in the opposite direction of the resultant of all the other forces added. To determine the equilibrant analytically, the resultant vector is first calculated using equation 2.2 below,
(Eq. 2.2)
where R is the resultant vector, R y is the summation of all vectors along the y-axis and R x is the summation of all vectors along the x-axis [1]. The magnitude or mass of the analytical equilibrant is equal to the magnitude of the resultant vector calculated using equation 2.2. The direction of the resultant is given by the following equation,
(Eq. 2.3)
where ϴ is the angle or direction of the resultant [1]. However, since the direction of the equilibrant is opposite that of the resultant vector, 180° is added to the angle calculated using equation 2.3 to get the angle of the equilibrant. The obtained values are indicated in Table 2.1.
Case 1 2 3 4 5
Table 2.1. Theoretical and Analytical Equilibrant Theoretical Analytical Mass (g) Position (°) Mass (g) Position (°) 200.80 210 202.71 210.96 199.81 210 232.03 210.41 221.61 210.81 200.88 208.33
% Error Mass 0.50 1.45 16.13 10.91 0.54
Position 0 0.46 0.07 0.39 0.80
The graphical vector addition was also used in the experiment to determine the equilibrant. Since there is a case (Case 5) which involves more than two vectors, the method used was the Triangle or Tail-to-Tip method, where the resultant is obtained by connecting the tail of the first vector to the head of the last vector. Since the equilibrant has the same magnitude but opposite the direction of the resultant, the resultant is extended towards the opposite direction until the same magnitude is obtained. The equilibrant values obtained using the graphical technique are in Table 2.2.
Case 1 2 3 4 5
Table 2.2. Theoretical and Graphical Equilibrant Theoretical Graphical Mass (g) Position (°) Mass (g) Position (°) 200 210 200 211 199.81 210 200 209 220 211 200 209
% Error Mass 0.10 0.10 0.10 10.10 0.10
Position 0 0.48 0.48 0.48 0.48
Based on the results of the experiment, the graphical method had less errors compared to the analytical technique, particularly in determining the magnitude of the equilibrant. However, theoretically, the analytical technique is more accurate because more errors may be committed while doing the graphical method. The values
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obtained in the equilibrants in the graphical method has a higher range of possible values compared to that obtained using the analytical method which only involves the range of error from the experiment. For the last portion of the experiment, a block of wood of unknown mass was used to replace hanging mass 1. In the first trial, two identical masses were added to the system and KTRATC was performed. A second trial with two identical masses but heavier than the first was conducted. To calculate the mass of the wood, it is assumed that the block of wood is the equilibrant. Since the equilibrant balances the other two hanging masses, its magnitude is therefore equal to the resultant of the two added hanging masses. The resultant mass can be obtained using equation 2.2. The values for the calculated and the actual mass of the block of wood are in Table 2.3 below.
Trial 1 2
Table 2.3. Unknown Mass Calculated Mass Actual Mass 133.54 115.92 114.53
% Error 15.20 1.20
As shown in Table 2.3, the first trial has a high percentage error of 15.20 % while the second trial has a low percentage error of 1.20 %. Since the hanging masses are identical, the errors may be explained by the angles at which the two masses were positioned. Theoretically, since the two masses are identical for each trial, the direction of their resultant must be directly opposite, or 180° away, to the equilibrant, which is the block of wood. To be able to do this, the two masses must be equally distant from the ideal resultant direction. However, in the experiment, they were not positioned this way while KTRATC was done. Again, the errors may also be partially due to the ignored friction between the strings and the pulleys.
Conclusion The main objective of this experiment is to use analytical, graphical and experimental techniques to verify the methods of vector algebra. The use of the tail-to-tip method was done for both graphical and experimental technique while the experimental technique was done with the use of a force table. Results for all cases had a percentage error below 15% so it is acceptable except for case 3 which had a 15% error which may be caused either by the changes in the level of the tale or the weights were not pr operly weighed. With the small error percentages determined in the results and discussion, it can be said that the results of the three techniques are close to one another. The data gathered experimentally is known to be precise relative to what was computed analytically and graphically. The methods of vector algebra using experimental, graphical, and analytical technique are thus verified. Since the unknown mass of the wood, as solved using the force table, is close to the value presented by the toploading balance, it can also be concluded that one could compute mass, and other different magnitudes of physical quantities, by using the principles of vector addition. With this experiment, it was confirmed that for KTRATC to happen, the magnitude (force) acting from it should be equal to zero; that the resultant vector must be equal to that of the equilibrant. The students recommend the future experimenters to first familiarize themselves with the concepts of vector addition before performing the activity. Instead of pure trial and error, they should consider doing the analytical technique along hand. Having a list of certain values and probabilities (i.e. where to position the pulley; ranges of mass of the objects to be hung) could help the experimenters finish their task on time.
References th
[1] Giancoli, D. (2014) Physics: principles with applications. 7 ed. Pearson Education: Upper River, N. J.
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[2] Young, H.D. & Freedman, R.A. (2008). University Physics with Modern Physics, 12th Ed. Pearson Education. Inc.: San Francisco.
