Hookes Law: Springs and Oscillators - Lab Report Example

Lab Report Hooke’s Law Springs and Oscillators This experiment investigates, preliminarily, the relationship between force and stretch as a stepping stone in the determination of the relationship between frequency and mass in an effort to determine the dynamic spring constant. As much as there exists a linear relationship between stretch and force according to Hooke’s law, this relationship becomes nonlinear when the object is oscillated at different speeds. The speed at which the mass is oscillated factors in the kinetic energy which plays a role in the nonlinearity. Introduction The fundamental objective of this experiment was to determine the key elements of Hooke’s law, which include, among others, the simple harmonic motions, the requirements of these harmonic motions, understanding Hooke’s law in a nutshell, attempting to verify Hooke’s law using a simple spring, to ascertain the relationship between the frequency and the period of a spring, to come up with the spring constant, and to determine the relationship between the mass, period, and force of a spring undergoing harmonic motion among others. The bottom line objective in this case is to be able to make comparisons of the measured periods of the vibration and the ideal ones as stated or calculated in the theory. Measuring the static behavior of a simple spring in this experiment was aimed at providing a basis of establishing the existing relationship between stretch and force. The spring constant, which is the force that a spring exerts on the object, in the opposite direction, when an object applies a force on the spring is what this experiment aimed at achieving. According to Hooke, the spring constant can be calculated from the expression F=- KX, where F is the force applied on the spring and X the distance this spring would move as a result of the force. In this experiment, a simple experiment involving a mass, a spring, and a meter rule were used to test this phenomenon (Knight, Brian and Stuart 13). Data and results The tables below show the results of the different measurements obtained as a result of adding different masses on a spring, while recording the various stretches that are realized. Concisely, the objective was to determine the influence of the mass on the stretch of the spring and the nature of relationship that exists between these tow variables. Traditional added mass Position 1 measured down (cm) Position 2 measured up (cm) Mg: felastic Displacement (cm) .01 67.4 67.5 0.1 0.1 .02 71.0 71.3 0.2 0.3 .03 75.0 75.1 0.3 0.1 .04 78.9 79.0 0.4 0.1 .05 82.6 82.8 0.5 0.6 .06 86.8 86.7 0.6 0.1 Averages (p1 +p0)/2 cm Traditional added mass Position 1 measured down Position 2 measured up Mg: felastic Displacement .01 58.5 61.4 0.1 2.9 .02 63.2 66.1 0.2 2.9 .03 68.0 71.5 0.3 3.5 .04 73.3 75.8 0.4 2.5 .05 77.8 79.4 0.5 1.6 .06 82.4 82.5 0.6 0.1 The above results give an impression of a near linear relationship between the mass and the stretch, but the trend changes significantly after a certain mass has been exceeded. From the theoretical point of view, the point where the relationship between the stretch and the mass of the objects tends to be nonlinear, the spring is said to have reached the elastic limit. In this case, plotting the graph of force against extension gives a near linear relationship between force and extension, just as far as the elastic limit. Due to possible experimental errors, for example, inaccurate reading of the meter rule, there is a likelihood of flier data points/values in the results. A section of the data plotted on the graph, suing the two set of data gives something close to the sketch below. The sketch is a graph of extension, in meters plotted against gravitational force. This will play a significant role in determining the dynamic spring constant by investigating the relationship between the frequency and the mass. Discussion Essentially, this experiment attempts to verify the linear relationships between the extension, or the displacement in this case with the difference force exerted by different objects. The general equation that governs the relationship between these variables is f (x) = kd x, taking kd as the dynamic spring constant at x0 = 0. The relationship between the aforementioned variables is not entirely linear as stated in theory. This is an indication that the equation f (x) = k x is not obeyed in this case. Looking at the values in the tables in the previous section, especially the second table, it can be realized that we can only investigate the differential spring constant, otherwise referred to as the local spring constant. Since this relationship is clearly non linear, we can only express this spring constant as a derivative, thus k local (x) = d F/dx . From this derivative, we can deduce then that the value of k local, or the local spring constant varies with stretch. Another noteworthy observation from the results, as well as from the theory as put forward by Hooke, is that any object that is subject to a linear restoring force will always undergo a simple harmonic motion when displaced and let go. Contextualizing the findings of this experiment with the second law of Newton, which states that F net = ma would give a differential equation such as d2 x/ dt2 = - kd x. The presentation of quality results as far as investigating the practicability of the theory is concerned is highly dependent on the method used or the experimental procedure. The experimental procedure has to be one that limits as much as possible. Errors in such experiments would result not only to an inaccurate value or poor estimate of the value under investigation, but would also cause inconsistencies in the results of subsequent results. The values in this experiment, however, have a significant similarity with the theory, as they demonstrate a somehow liner relationship when the focus is on the stretch and the mass. This relationship tends to be nonlinear when the spring is oscillated (Knight, Brian and Stuart 13). The contributing factor to the nonlinearity of the local spring constant is the fact that any spring has elastic potential energy that is always proportional to the stretched distance squared, or even the when compressed away from rest or its equilibrium. The objects we attached on the spring contain have kinetic energy which is always proportional to the speed at which that spring or object moves. The kinetic energy is expressed as KE = ½ MV2. Determining where the potential and kinetic energies are at their minimum and maximum was key to the determination of the variables under investigation in this experiment. Works Cited Top of Form Knight, Randall D, Brian Jones, and Stuart Field. College Physics: A Strategic Approach. Boston: Addison-Wesley/Pearson, 2010. Print. Bottom of Form Read More