Lab Addition of Vector Forces

Addition of Vector Forces ABSTRACT Vectors are commonly used for different forces by indicating size and direction of the force. There are two common ways to add vectors in order to determine the resultant—the graphical method and the component method. When the forces sum is zero, an object is in equilibrium. This experiment used the two methods to demonstrate a system in equilibrium. Three or four weights and angles were used in order to determine the varying vectors required to center a small metal ring. Error range and sources of error for the equilibrating forces was determined. The resultant vectors were determined by both graphical and component methods. The experiment supported the theory of equilibrium because the vector fell in the appropriate range. THEORY Vectors are commonly used to indicate the size and direction of different forces. The magnitude of the vector can be determined mathematically by using the Pythagorean theorem The angle with which the vector makes relative to the x axis will be used to determine the direction of the vector and can be found using The sum of two vectors is called a resultant, and there are two common methods in order to find a resultant: the graphical and component methods. The graphical method involves precicely drawing the vectors using rulers and protractors on graph paper. Two vectors are added together by placing the tail of the second vector at the head of the first vector. The resultant is then measured from the tail of the first vector to the head of the second. This can be easily seen with Figure 1. Figure 1. Vectors a and b are added together to form vector c. The component method is achieved by adding the axis projections of each of the vectors together. The Pythagorean theorem is then used to determine the resultant (R). R can be calculated with the following equation making sure to separate the vectors into components of x and y using trigonometry. The angle of the resultant vector can then be calculated by Vector addition can be used to sum forces and determine whether they are in equilibrium (vector sum of forces is zero). This can be shown by either method of vector addition; however, they can also be used to show if vectors are in equilibrium. The following equations both demonstrate equilibrating forces, one by setting the vector sum equal to zero and another by showing the resultant force to be equal in magnitude, but opposite in direction of vectors. OBJECTIVE The purpose of this lab is to use vector addition by graphical and component methods in order to show equilibrating forces. PROCEDURE Two unequal masses were placed on two different strings, one and two. Another string was also used, and the three strings were allowed to hang over the force table at three different angles. Using a trial and error technique, mass was added to string three and the angle was changed in order to find the point of equilibrium. Both the graphical and component methods discussed were then used to demonstrate that the system was in equilibrium. The resultant of vectors one and two was calculated. Additional mass was added to one of the strings until the ring at the center of the setup was touching a post on the side. Then mass was removed from that same string until the ring touched the opposite side. The same procedure was then repeated for a ring with four strings attached to it. Experimental set-up below: Figure 2. Illustration of the experimental set-up. DATA Part 1 Mass Angle (degrees) (g) Mass high (g) Mass low (g) 64.8 150 150 160 287.5 235 255 217 Part 2 Mass Angle (degrees) (g) 64.8 150 150 160 240 190 342.1 190 CALCULATIONS F=mg 1.47N= .150kg * 9.8m/s/s √ (− 0.1 N )2+( 0.3 N )2 = 0.3N Xn = ‖ R‖cosθn 0.3N(cos287.5o) = -0.1N R‖ sinθm Ym = ‖ 0.3N(sin287.5o) = 0.3N ΔF= M h−M l g 2 186.2N = (255 g−217 g ) (9.8m/s/s) 2 QUALITATIVE ERROR ANALYSIS The data collected for this lab may not have been entirely accurate due to the friction from the strings and pulley system. For this experiment, the force of friction was ignored in calculation and measurements, but it does play a role in the net equilibrium of forces. Another area of error could have been due to inaccuracy of drawing vectors. Since this was done by hand, there is a possibility of error in vector length and angle accuracy. The masses used for the experiment were also assumed to be accurate; a “50 g” mass was assumed to be 50 grams, while it may have been 51 grams. QUANTITATIVE ERROR ANALYSIS During this experiment several measurements were made. Among them were the angle measurements. These measurements were made with uncertainty in n±0.5o. The error of the mass for part one had a range of 217g≤235g≤255g. This range means that the error within the equilibrating mass could have this range. The error range for the experiment was 0.3N+2(186.2N)≥0.3N≥ 0.3N-2(186.2N). RESULTS Force Graphical (N) Resultant (N) 1.47 1.57 2.31 0.3 Graphical Force (N) Resultant (N) 1.47 1.57 1.86 1.86 0.15 Part 1 Component Resultant (N) X Component (N) Y Component (N) 0.3 -0.1 0.3 Part 2 Component Resultant (N) X Component (N) Y Component (N) 0.15 -0.14 0.49 CONCLUSION The equilibrium theory was supported by this experiment. The error range was 0.3N+2(186.2N)≥0.3N≥ 0.3N-2(186.2N), so the experimental value was within that of the expected theoretical value. The resultant vectors and equilibrium vectors were small in magnitude relative to the other forces. Since the sum of the forces should equal zero in order to be in equilibrium, the experiment showed that the system was close to it.