A Sample Lab Report Exp 0 Empirical Equations

A Sample Lab Reportexp 0 Empirical Equations

A study was done of the relationship between the diameters of various sized rings and their natural periods of oscillation when allowed to swing as a pendulum. Using the acquired data and the general form of the equation of a pendulum, , which is of the form: , values for the proportionality constant, A, and the power, n, were empirically determined. The numerical value for A was determined to be equal to 0.196, and a numerical value for n was determined to be 0.509. A comparison was done for the values of these same constants to that of a simple pendulum. A percent difference of 2.3% was determined for A and a percent difference of 1.8% was determined for n with this comparison. From this it is inferred that the equation for the period of a Ring Pendulum as a function of its diameter is of the same form as that for the period of a Simple Pendulum as a function of its length.

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Introduction

The study of pendulums has historically played a significant role in understanding fundamental principles of physics, especially in the realm of oscillatory motion. Pendulums, both simple and complex forms such as the ring pendulum, serve as essential tools for experimental physics owing to their predictable behavior governed by gravitational forces and geometric parameters. This research aims to empirically determine the relationship between the diameters of rings and their oscillation periods, thereby extending the theoretical understanding of pendulum motion and verifying the applicability of simple harmonic motion equations to ring pendulums.

The core premise is that the period of oscillation, T, for a ring pendulum might depend on its diameter, d, following a power-law relationship of the form T = A * d^n, where A is a proportionality constant and n is an exponent that indicates the nature of this dependence. By measuring the oscillation periods of rings with different diameters and plotting these data on a logarithmic scale, the experiment seeks to validate or refute this hypothesis and compare the empirical constants with those derived from simple pendulum theory.

The empirical approach is instrumental here because it focuses on observed data rather than assumptions rooted solely in theoretical derivations. Holding variables such as mass and material properties constant, varying only the diameter, allows for clear examination of how the geometric parameter influences oscillatory behavior. The expected outcome is that the period scales with diameter in a specific, predictable manner, thereby aligning the physics of ring pendulums with established principles of simple harmonic motion.

Methodology

The experiment involved measuring the oscillation periods of five rings with varying diameters. Precise measurements of inner and outer diameters were taken using Vernier calipers for smaller rings and a meter stick for larger ones to minimize measurement errors. Each ring's mean diameter was calculated from multiple measurements to enhance accuracy. To determine the period, each ring was displaced slightly and its oscillation recorded over multiple cycles. The average period was then calculated to account for experimental variability.

Data analysis centered on plotting the logarithm of the measured period (log T) against the logarithm of the mean diameter (log d), enabling linear regression to determine the slope and intercept. The slope corresponds to the exponent n in the power-law relationship, while the y-intercept yields log A, from which A is derived. The plot on a log-log scale makes it easier to identify linear relationships and confirm the power-law dependency.

Additionally, a direct plot of T versus d on a standard scale was performed, expecting a nonlinear relationship that supports the power law identified in the log-log analysis. These data were analyzed using Excel's trendline tools, providing empirical values for A and n along with percent error calculations relative to theoretical expectations based on simple pendulum equations.

Results and Discussion

The analysis yielded an empirical value of the proportionality constant A as approximately 0.196 and the exponent n as approximately 0.509. When comparing these values with those obtained from a simple pendulum, where theoretical models suggest that the period depends primarily on length L and gravitational acceleration g (T = 2π√(L/g)), a reasonable analogy can be drawn. Since diameter d is proportional to some effective length parameter in the ring's oscillation, the empirical values closely approximate those anticipated by theoretical models.

Percent differences were calculated: A showed a discrepancy of about 2.3%, which indicates a high degree of correlation between empirical data and theory, suggesting that the initial assumption of a power-law relationship is valid within experimental uncertainties. Similarly, the difference in n was approximately 1.8%, reinforcing the hypothesis that the period scales with the diameter raised to a near-half power.

The small discrepancies could derive from measurement uncertainties, assumptions about uniformity in ring material properties, or approximations in data processing. Notably, the results support the concept that the pendulum's period dependence on diameter mirrors theoretical expectations derived from the physics of simple pendulums, with the primary difference being in geometrical factors, which are appropriately encapsulated by the empirically determined constants.

Conclusion

This experiment confirms that the period of a ring pendulum exhibits a power-law dependence on its diameter, with empirical constants closely aligning with those predicted by simple pendulum theory. The findings support the hypothesis that, despite differences in geometry, ring pendulums obey similar oscillatory principles as simple pendulums when considering their effective length or diameter as a key parameter.

The close agreement between the empirical constants and theoretical expectations demonstrates the validity of using an empirical approach to explore complex physical relationships where direct theoretical derivations might be cumbersome. Future studies could deepen this analysis by considering additional variables such as ring material density and thickness to refine the empirical model further.

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