Exploring Oscillatory Energy Dynamics: Validating Conservation Laws in Mass-Spring Systems

The objective of this laboratory experiment is to validate the law of conservation of energy by measuring the potential and kinetic energies at different positions of an oscillating mass connected to a spring.

The sum of these energies is compared at various points to confirm their equivalence.

The experimental setup involves a spring attached to a metal rod with a mass (0.5005 kg) at one end, along with a motion detector. The oscillation is initiated by pulling the spring down and releasing it, causing vertical motion.

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The motion detector records the position of the mass over time.

The collected data is represented in a graph, showing a sine wave indicating the oscillation frequency of approximately 8.08 rad/s. A period with minimal noise is selected, and the corresponding set of slopes (velocity of the mass) during this period is determined.

Three situations are discussed in free-body diagrams representing the mass oscillating at different points: (a) equilibrium or the middle of the oscillation, (b) when the mass accelerates upward at the bottom of the oscillation, and (c) when the mass accelerates downward at the top of the oscillation.

The equations for potential and kinetic energy at the top, middle, and bottom positions are provided, taking the position where the spring is neither stretched nor compressed as the reference point for potential energy. These equations are used to analyze the energy at different stages of the oscillation and validate the law of conservation of energy.

Objective: The primary goal of this laboratory experiment is to validate the law of conservation of energy by measuring potential and kinetic energies at various positions of an oscillating mass connected to a spring. The sum of these energies is analyzed to determine if it remains constant at different points in the oscillation.

Procedure: The experimental setup involves a spring hanging from a metal rod with a mass of 0.5005 kg attached to one end. A motion detector is used to record the position of the mass over time. The oscillation is initiated by pulling the spring down and releasing it, causing vertical motion.

Data Collection and Analysis: The position of the mass concerning time is recorded and plotted on a graph, revealing a sine wave with a frequency of approximately 8.08 rad/s. To minimize noise, a specific period is selected, and the corresponding set of slopes (velocity of the mass) during this period is determined.

Free-Body Diagrams: Free-body diagrams for three situations - equilibrium, upward acceleration, and downward acceleration - are presented. These diagrams assist in understanding the forces acting on the mass in different phases of the oscillation.

Energy Equations: Using the position where the spring is neither stretched nor compressed as the reference point for potential energy (Us = 0), three sets of equations are derived for potential (Ug), kinetic (K), and total mechanical (E) energy at the top, middle, and bottom positions.

Total Mechanical Energy Calculation: The total mechanical energy for each situation is calculated using the derived equations. This involves substituting the acquired data into the formulas to obtain values for kinetic, potential, and mechanical energy at each position.

Results and Discussion: The obtained data is presented in a table showcasing kinetic, potential, and mechanical energies at each position of the mass. A graphical representation of kinetic, potential, and mechanical energies as functions of time is also provided.

Analysis of Results: Upon analyzing the data, it is observed that the average mechanical energy of the mass is 0.27 ± 0.3J. Theoretically, mechanical energy should remain constant at every point in the oscillation. However, the data points in the graph do not align in a straight line, and the fractional uncertainty is not within the preferred range (smaller than 0.1). This deviation may be attributed to noise from the motion detector and potential human errors.

Conclusion: In conclusion, the experiment aimed to confirm the law of conservation of energy in an oscillating mass-spring system. While the theoretical expectation is constant mechanical energy at every point, the observed data deviates, possibly due to experimental limitations. Further refinements in data collection and reduction of noise may enhance the accuracy of the results. Despite the current discrepancies, the experiment contributes to the understanding of energy conservation principles in oscillatory systems.