Dimensional Analysis: Investigation on bifilar suspension
Ho Yuk Him (Date of experimentation :20th and 29th October, 2010) This paper presents a study on the application of dimensional analysis on the formulation of an equation concerning the periodic time of a rod rotating at an angle about the vertical axis suspended by two strings at each end. We find that period is inversely related to distance between the strings parallel to each other and directly proportional to the both the square root of length of string and the length of suspension, (i.e. ｄ
). Although discrepancy is found when the equation proposed
by eminent scholars is compared against with ours, our findings can hardly pale into insignificance as errors and the cause of uncertainty are discussed and ways to improve the experiment are suggested, which can possibly spur further studies on this realm of knowledge.
I. Introduction In physics, there are mainly two ways from which useful equations can be derived. One from them is to make good use of the mathematical expression of the theories formulated by our predecessors. An archetype to this notion is the Newton’s second law, which states without ambiguity that “ The rate of change of momentum of an object is directly proportional to the net force acted on it, and the motion occurs along the direction of the force”. This statement can be paraphrased using mathematical terms, i.e. , (1) where F denotes force, m mass and a acceleration. This equation has a formative impact on the realm of physics, as it helps the derivation of other equations. For instance, answering a question concerning the force acting on a ball of mass m being in a circular motion requires the use of (1). Given the centripetal acceleration, , (2) where a denotes acceleration, v velocity and angular velocity, no additional equation is needed, as (1) suffices the requirement. Obviously, the force acting on the ball in its circular path is, . (3) Encouragingly, this method has yielded fruitful outcomes and it continuously help explain phenomena considered inexplicable by our ancestors. However, all methods have their own limitation. The method suggested above can neither escape from its doom. It is not workable to equations that require enormous back-ups or when the “prerequisites” for their formulation simply do not exist at the time of investigation. It is, therefore, necessary to tackle the problem by deploying another more versatile method, the basic principle in dimensional analysis. In fact, dimensional analysis is a technique used both to check the validity and to find the relationship between physical quantities. The basic principle of it is that “for two quantities to be equal, they must have the same units or convertibly, same dimension.” A terminology is used to describe the state in which the quantities on both sides of the equation carry the same unit and it is “Homogeneity”. An equation is said to be homogeneous when it satisfies the requisite stated above. Its use is explained in great details in the following sections. We first discuss what apparatus are used, the set-up of our experiment along with the hypotheses we made
before experimentation, under the sub-heading background. Then, equations and rules considered useful to our experimentation and details of how the experiment is conducted are presented under methods, which is subdivided into theory and experiment. It is immediately followed by the section concerning the results obtained. Under results, data tabular form is given together with a detailed list of procedures on how figures arrive. Their respective errors and uncertainties falling into the category random error are also given. However, graphs and sources of random and systematic error, which can hardly be detected, are presented in the section analysis. After that, discussion and conclusion will impart readers of the discrepancy between the standard result found by other scientists and the result from...
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