The Equilibrium States in Vector Mechanics - Statics and Dynamics - Lab Report Example

Vectors and Statics Partner’s Group Number TA’s Theoretical Background In physics of vectors and statics, an object is said to experience balanced forces whenever; the net force is equal to zero, and acceleration is equal to 0 ms-2. Objects experiencing balanced forces could either be at rest or moving at uniform speeds towards the same direction. This experiment involved the study of balanced objects at rest; otherwise known as objects in a static equilibrium state. Technically, static objects experience angular forces which maintain the objects’ resting position. When all the forces acting on a stationary object are summed together as vectors, then the resultant force must be equal to zero (Beer 62). Practically, determining a static object’s net force involves either drawing an accurate vector summation diagram and adding all the individual forces, or using trigonometric functions to convert individual forces into their vertical and horizontal components; hence comparing whether vertical and horizontal force components are balanced. Objectives The main objective of this experiment is to employ trigonometric functions in resolving the net force of an object at static equilibrium. In theory, vertical forces are determined using the sine rule whereas horizontal forces involve the use of cosine rule. Primarily, a suspended object exerts tension on the suspender string. In addition, the suspended object exerts weight due to gravitational acceleration. By suspending an object at an angle, the string’s tension can be used to determine the magnitude and direction of forces. Besides the use of trigonometric functions, this experiment also sought to practically determine the magnitude and effects of experimental errors in laboratory exercises. In vectors, errors propagate in two fronts; forces’ magnitude and direction (Beer 70). Therefore, another objective of this experiment is to examine the manner in which errors propagate through both angles and magnitude of component forces. Finally, this experiment provides a means through which sources of experimental errors can be predicted and explained in a contextual manner. Experimental Data Data obtained from each procedure of the pulley experiment were tabulated below. The tabulated data include masses of weights, angle of tensile strings, and forces exerted by each weight. Mass (g) Angles (in radians) Forces (N) M1, 100.3 30o = 0.524 radians T1 = 9.830 M2, 150.15 50o = 0.8727 radians T2 = 14.715 M3, 162.64 86o = 1.5001 radians T3 = 15.939 Data Analysis In this section, forces and angles tabulated above were used to deduce the magnitude of each force, directional components of each force, the net force experienced by the static object, and errors propagated through magnitude and direction parameters. Vectors of Each Force Formula for finding the magnitude of each force is given by; Fx = F cos θ and Fy = F sin θ where x and y are the directional components of each force in Cartesian coordinates (Beer 70). Vector components of each force plus their summations are tabulated below. Force (N) Conversions Magnitude of each force (N) T1 9.830 Tx1 = 9.830 cos 30o/9.830 cos 0.5240 radians 8.510 T1 9.830 Ty1 = 9.830 sin 30o/9.830 sin 0.524 radians 4.915 T2 14.715 Tx2 = 14.715 cos 50o/14.715 cos 0.8727 radians 9.459 T2 14.715 Ty2 = 14.715 sin 0.8727 radians 11.272 T3 15.939 Tx3 = 15.939 cos 86o/15.939 cos 1.5001 radians 1.112 T3 15.939 TY3 = 15.939 sin 86o/15.939 sin 1.5001 radians 15.90 Summation of Tx Tx = 8.510 - 9.459 + 1.112 = 0.1630 N Summation of Ty Ty = 4.915 + 11.272 – 15.90 = 0.287 N Net Force T = √∑ (Tx2 + Ty2) T = √(0.1632 + 0.2872) T = 0.3301 N Propagation of Errors As acknowledged earlier, errors propagated through angles and magnitudes of each force. As a means of enhancing confidence and certainty of the results obtained in the conversions above, the margin of error in each of the three forces was calculated (Beer 73). Errors of each force was calculated and tabulated below. Force Conversion formula Calculation Error TX + Ty + T Results After calculating vector components of each force together with their respective errors, the following tabulated results were obtained; Vector (Newton) Certainty range Tx = 0.1630 N Tx ≤ 0.1630│79.55│ Ty = 0.287 N Ty ≤ 0.2870│39.44│ T = 0.3301 N T ≤ 0.3301 │88.79│ Discussion and Conclusion As aforementioned, the main objectives of the experiment included; the application of trigonometric functions in deducing vectors, determination of errors propagated through vectors, and finally acknowledging the sources and effects of those errors in vectors and statics experiments. First, trigonometric functions of sine and cosine rules were instrumental in facilitating conversion of the scalar parameters into vector forces. For example, the scalar parameter M1 which weighed 100.3 g was converted into the vector parameter T1; 9.83 N. In addition, the trigonometric functions enabled deduction of individual directional forces acting on the static object. For example, T1 was divided into Tx1 and Ty1, thus determining the vertical and horizontal forces acting on the object. At this juncture, it emerges that vertical and horizontal forces were required in order to establish static equilibrium on the suspended object (Beer 73). From the experiment’s results, it emerged that a horizontal force of 0.1630 N and a vertical force of 0.2870 N were exerted in order to balance the object. Admittedly, horizontal forces and vertical forces must not be equal in order for an object to achieve static equilibrium. At equilibrium, the static object was supposed to experience net forces of 0 Newton from both the vertical and horizontal forces. In addition, the net force, T, for the object was supposed to be 0 N (Beer 74). From the experiment’s calculations, all the three forces failed to meet the 0 N requirements. This incongruence can be attributed to errors propagated through magnitude and direction parameters during calculations. Therefore, calculations pertaining to the propagation of errors were performed. These calculations provided a certainty range under which the actual value of each directional force can be found. For example, the actual value of the horizontal force is Tx ≤ 0.1630│79.55│. In this context, Tx = 0.1630 ± 79.55 N. Similarly, errors in both the vertical and net forces were acknowledged in form of confidence ranges. Thus Ty ≤ 0.2870│39.44│ whereas T ≤ 0.3301 │88.79│. Errors that infiltrated into the experiment’s data and results were either random or systematic in nature. One probable cause of random errors in the experiment was instrument resolution. Scales used in measuring mass of the suspended object and angles of tensile strings had limited precision. In this context, the inability to obtain precise measurements of scales smallest than 0.1 g in mass and 0.1o in angles were the primary source of the experiment’s random errors. Contrarily, parallax was a potential source of systematic errors in the experiment. In obtaining instrument readings, observers must position their eye neither higher nor lower from a scale’s pointer. Technically, failure to squarely align one’s eyes while obtaining scale readings causes significant errors. Therefore, inconsistencies propagated in the vectors and statics experiment can be attributed to the discussed systematic and random sources of errors (Beer 75). In conclusion, it emerged that comprehension of vectors and statics is essentially necessary in physics classrooms. Vectors and statics in equilibrium is a basic concept in the physical aspects of engineering and construction (Beer 76). Undeniably, this experiment enhanced the ability to discern both the magnitude and direction of forces acting on a static object. In this regard, the experiment’s objectives were covered successfully. Work Cited Beer, Ferdinand. Equilibrium States in Vector Mechanics: Statics and Dynamics. New York, NY: McGraw-Hill Companies, 2010. Print. Read More