Physics lab
Force Table Lab Partners: Person 1, Person 2, Person 3, etc. Instructor, T.A.: Your Instructor, Your TA MM/DD/YY ABSTRACT This experiment was conducted to show how vectors affect one another- in particular, how opposing vectors can be added up to cancel each other out to create a system in equilibrium, which was done by hanging different masses over various angles on a force table. As a result, each case showed that when summed all forces added to 0. INTRODUCTION Vectors are extremely important in physics, as they provide a way to show quantity that has not only a magnitude, but a direction as well, which is extremely important when explaining things like motion. Although these vectors are more complex than just a single number, they can be manipulated by other vectors fairly easily. This makes combining certain measurements that could involve a multitude of vectors, as well as manipulating a single vector as it can be added or subtracted from itself, fairly simple. This experiment showed the use of a force table to prove this manipulability with vectors by setting mass as forces on certain angles in order to cancel each other out. This works as an example because all three of the masses had some sort of force, in this case being caused by acceleration due to gravity, being applied to them in the direction they were angled. It also helped to demonstrate graphical methods for manipulating vectors by means of “tip-to-tail” measurement. This type of measurement aids in the visual representation of vectors and gives understanding to how a system of vectors looks when in equilibrium, in this case a quadrilateral formed by four vectors of different magnitude and direction. A number of equations were used in this experiment, and are as follows: instructor name F x = 0Σ (1) F y = 0Σ (2) F x = Fcos( )θ (3) F y = Fsin( )θ (4) g = 9.8 m/s 2 (5) F = mg (6) Equations (1) and (2) show how F x and F y , the horizontal and vertical components of force F ( Newtons ), when in an equilibrium-system should sum to 0. Equations (3) and (4) show how the force F is geometrically related to the horizontal and vertical components, respectively, by means of angle ( degrees ). Equation (5) is a constant that states how the acceleration due toθ gravity, g ( meters/second 2 ), is equal to 9.81. Equation (6) is a variation of Newton’s Second Law that shows that the force due to gravity on an object is equivalent to g multiplied by mass m ( kilograms ). PROCEDURE The force table, which allows a central equilibrium to be reached by hanging multiple masses at different angles, was set up with 3 points to be determined. The force table with a 3-pulley setup is seen in Figure 1 . The pulleys were attached around the circumference with a ring and three strings that could spin freely placed in the center of the table. The first trial included forces in the first quadrant at 0 o degrees and 200 grams and 90 o degrees and 400 grams, and one component to be measured based on these. The third string was then pulled across the table until the ring was centered on the middle post, and the angle was recorded. A mass hanger was then added to the string on this side, and weight was added until the ring again centered itself in the middle. At this point, all components were put in equilibrium. This process was repeated once more, only the 90 0 angle was adjusted to 135 o with the same weight of 400 grams. The third angle was measured, weight was added, and equilibrium was reached, and the results were recorded. Following this, a procedure to allow for a graphically calculated equilibrium-system was conducted. An arbitrary quadrilateral, with a base line on the horizontal of some graph paper, was drawn. Each side represented a tip-to-tail vector, with the final tip touching the tail of the horizontal vector. The angle of each vector was found using a protractor and the magnitude of each was found with a ruler using the conversion 0.25” = 25 grams. These values were translated to the force table, where 4 hangers and pulleys were used to represent the four vectors in the quadrilateral. Small adjustments were made until equilibrium was established. RESULTS AND CALCULATIONS The following table shows angles, mass, and force for the first setup in the experiment, which involved starting angles that were orthogonal. Following the table are sample calculations demonstrating how the results were found. Table 1: Orthogonal Starting Forces, Resultant Force Angle ( o ) Mass (kg) Force (N) 0 0.200 1.96 90 0.400 3.92 246 0.450 4.41 F = mg (6) g = 9.8 m/s 2 (5) 450 g 10 -3 = 0.450 kg× 0.450 kg 9.8 m/s 2 = 4.41 kg = 4.41 N× m s 2 A table for the vertical and horizontal components is shown below, with sample calculations following: Table 2: Orthogonal Starting Forces, Components Angle ( o ) Force (N) X-component (N) Y-component (N) 0 1.96 0.200 0.000 90 3.92 0.000 0.400 246 4.41 -0.183 -0.411 Sum 0.017 -0.011 F x = Fcos( )θ (3) F x = Fcos( ) = (.450)cos(246) = -0.183 kg θ F y = Fsin( )θ (4) v y = vsin( ) = (.450)sin(246) = -0.411 kg θ F x = 0Σ (1) F x = 0.200 + 0.000 + -0 .183 = 0.017 0Σ ≈ F y = 0Σ (2) F y = 0.000 + 0.400 + -0 .411 = -0.011 0Σ ≈ The following table shows angles, mass, and force for the second setup in the experiment, which involved starting angles that were in the first and second quadrants. Table 3: First and Second Quadrant Forces, Resultant Force Angle ( o ) Mass (kg) Force (N) 0 0.200 1.96 135 0.400 3.92 285 0.330 3.23 A table for the vertical and horizontal components in this section of the experiment is shown below. Table 4: First and Second Quadrant Forces, Components Angle ( o ) Force (N) X-component (N) Y-component (N) 0 1.96 0.200 0.000 135 3.92 -0.283 0.283 285 3.23 .085 -0.319 Sum 0.002 -0.036 The following page consists of the graphs of Force and components for each of the two trials. They are based on the data in Tables 1 , 2 , 3 , and 4 : The following table shows the angles, magnitudes, and forces for the quadrilateral system of vectors that was drawn on the graph paper. Again, the magnitudes are based on 0.25” = 25 g. Table 5: Quadrilateral System, Forces Angle ( o ) Mass (kg) Force (N) 0 0.25 2.4525 116.57 0.1118 1.0968 180 0.15 1.4715 243.43 0.1118 1.0968 A table for the vertical and horizontal components in this section of the experiment is shown below. This is the data the was calculated before being put on the force table. Table 6: Quadrilateral System, Components, Theoretical Angle ( o ) Force (N) X-component (N) Y-component (N) 0 2.4525 2.4525 0 116.57 1.0968 -0.4905 0.9810 180 1.4715 -1.4715 0 243.43 1.0968 -0.4905 -0.9810 Sum 0 0 DISCUSSION, DATA ANALYSIS, AND CONCLUSION In total, the data was fairly accurate in portraying systems in equilibrium. Each case showed fairly close results, meaning each mass system tested ended with the ring being centered. In the first setup, where the starting angles were orthogonal, the resultant force was in the quadrant opposite, which is expected. When considering the vectors in terms of parallelograms, the resultant of those two forces would be on a line somewhere in between them, which when applied to create equilibrium would be in the opposite direction, For the case where the angles were in different quadrants, the resultant was more difficult to predict. In the case of creating a resultant force that will put a two-force system into equilibrium, it can be judged that the applied resultant force will tend to be directed more in parallel to the strongest of the two forces already in the system; in other words, the angle between the resultant vector and the strongest pre-existing vector will be the closest to 180 degrees. This proved true in the case of this experiment, as the third force applied created a much more obtuse angle with the hanger producing a greater force out of the two forces already on the table. The quadrilateral equilibrium system was a way to demonstrate differences in tip-to-tail/graphical versus component methods of vector computation. As the data shows, while graphical representation can be useful for visualization, component methods give much better, more precise theoretical values that are better suited for further calculation as these attributes have much less error. They are also more easily organized and are more efficient to use, which is why they are preferable. Like the two previous sections of the experiment, the quadrilateral that was conceived demonstrated a system of equilibrium; however, it gave more rise to sources of error due to the discrepancies and variations between the computed model and the force table model. One possible source of error that created some of these unexpected results was that the angles were all measured by eye. Obviously depending on the perspective each angle can look different, so this could have resulted in some inaccuracies when measuring not only the components, but the masses as the qualifications for equilibrium would have changed. Another source of error is that the force table had not been calibrated. Similar to the source above, this error may have caused an imbalance in the equilibrium, thus producing inaccurate component measurements. Another source of error may have been the strings on the center ring, and the friction in the pulleys. It was noted after the experiment that they were not freely sliding around as they should and were sometimes catching on certain portions of the ring. This was also the case with the pulleys. This means that some of the results may be off as the angle may not have been accurately measured after adding mass to the first two strings. Overall, this experiment showed how important vectors are and how their angles and magnitudes significantly affect their behavior and interaction with one another. It demonstrated not only how to solve equilibrium systems, through components as well as graphical representation, but how to calculate, create, and check equilibrium systems.
