# Pendulum Experiment: Exploring Factors Influencing Period and Small-Angle Approximation Validity

Categories: Physics

This laboratory experiment focuses on the study of a Simple Pendulum by analyzing its oscillation period through variations in mass, string length, and initial oscillation angle. By manipulating these factors, the dependence of the oscillation period is explored. The findings reveal that the period is solely influenced by the length of the string.

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No significant changes were observed in the period concerning different masses or angles, indicating a proportional relationship between the period and the square root of the string length.

Purpose: The Simple Pendulum, consisting of a mass suspended from an inelastic string, serves as a tool to examine oscillation patterns when subjected to alterations in various physical parameters.

This experiment aims to investigate the impact of changing these aspects on the pendulum's oscillation.

Theory: A Simple Pendulum, comprised of a mass attached to an inelastic string hanging from a fixed point, exhibits periodic motion when displaced from equilibrium. The time required for one complete oscillation, termed the period (T), is determined by the acceleration due to gravity and the length of the string. This relationship is valid for small-angle oscillations, portraying simple harmonic motion when the amplitude remains small.

Part 1: Exploration of Mass Influence Procedure: Three pendulum bobs with different mass compositions were chosen. These bobs were affixed to a constant-length string to assess the potential impact of mass on the measured period. Utilizing a photogate, the time for each bob to complete one oscillation was recorded through five trials for each mass.

Data:

Length of string (same for all trials): L=0.36m

Initial angle of Oscillation (same for all trials): ϴ=5⁰

 Mass (kg) Trial 1 (s) Trial 2 (s) Trial 3 (s) Trial 4 (s) Trial 5 (s) 0.0742 1.1978 1.1714 1.1945 1.1981 1.1818 0.2190 1.1961 1.1746 1.1881 1.1823 1.1961 0.2281 1.1552 1.1901 1.1910 1.1964 1.1985
 Mass 1 Mass 2 Mass 3 Average Time (s) 1.1887 s 1.1874 s 1.1862 s Maximum Error 0.0094 s 0.0087 s 0.0123 s Minimum Error 0.0173 s 0.0128 s 0.0310 s

Discussion: As per the previously provided equation for the period of oscillation, mass is not anticipated to play a role.

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Consistent with class notes, no mass dependence is expected, and the results affirm this expectation. The observed period variations are within a range of less than 1% of the measured values.

Part 2: Length vs. Time Procedure: A pendulum was assembled using the smallest bob employed in the first part. The oscillation period was recalculated by varying the length of the string. Measurements were taken from the top of the pendulum to the center of the bob using a standard ruler. Five trials were conducted for each distinct length.

Data:

Steel bob used: m=0.0233 kg

Angle of Oscillation: ϴ=5⁰

 Length (m) Trial 1 (s) Trial 2 (s) Trial 3 (s) Trial 4 (s) Trial 5 (s) 0.4400 1.3242 1.3272 1.3282 1.3286 1.3247 0.3700 1.2186 1.2186 1.2194 1.2168 1.2183 0.3200 1.1416 1.1443 1.1376 1.1387 1.1407 0.2900 1.0970 1.0961 1.0971 1.0971 1.0933 0.1940 0.8643 0.8712 0.8837 0.8642 0.8621
 Average Oscillation (s) Maximum Oscillation (s) Minimum Oscillation (s) 1.3266 0.0020 0.0024 1.2183 0.0011 0.0015 1.1406 0.0037 0.0030 1.0961 0.0010 0.0028 0.8691 0.0146 0.0070

The graphs illustrating Period vs. Length and Period vs. Square Root of Length demonstrate a clear pattern: the period increases with an increase in length. As outlined in the Theory section, the period of a Simple Pendulum is expressed as

Thus, the expectation was for T to rely on rather than L. The period was anticipated to be proportional to , with the slope of the linear relationship representing the factor in front of . The Expected Slope is calculated as:

Expected Slope=g​2π​

The Measured Slope, determined from the data, is 0.4878. This results in a Percent Error of 63.4%. Such discrepancies may arise from inherent errors in time measurement, length measurement (precision of the ruler), and angle measurement.

Part 3: Time vs. Angle Procedure: The small-angle approximation assumption, crucial in deriving Simple Pendulum equations, was tested. The oscillation period, utilizing the weight from Part 2, was calculated while varying the angle of displacement. Ten trials were conducted, each with a different angle of displacement.

length of String (Used through all trials): L=0.44m

Mass of Weight (Used Through all trials): m=0.02333kg

 ϴ (⁰) Trial 1 Trial 2 Trial 3 7 1.3361 1.3418 1.3365 5 1.3272 1.3286 1.3286 10 1.3396 1.3370 1.3410 15 1.3435 1.3423 1.3494 20 1.3464 1.3506 1.3496 25 1.3540 1.3548 1.3537 35 1.3788 1.3770 1.3775 40 1.3859 1.3829 1.3842 45 1.3949 1.3955 1.4014 30 1.3625 1.3599 1.3622
 ϴ (⁰) Average Period (s) Maximum (s) Minimum (s) 7 1.3381 0.0037 0.002 5 1.3281 0.0005 0.0009 10 1.3392 0.0018 0.0022 15 1.3451 0.0043 0.0028 20 1.3489 0.0020 0.0025 25 1.3542 0.0006 0.0005 35 1.3778 0.0010 0.0008 40 1.3843 0.0016 0.0014 45 1.3973 0.0041 0.0024 30 1.3615 0.0010 0.0016

The outcomes indicate that for angles less than 10⁰, the small-angle approximation holds true for Simple Pendulums, demonstrating that sinϴ is approximately equal to ϴ, as discussed in class. This approximation is valid when the angle is sufficiently small, effectively approaching zero.

Despite its simplicity, this lab provided an effective means to investigate the period of a simple pendulum. Experimental findings from Part 1 and Part 3 confirm that the pendulum's period is independent of its mass or angle. Additionally, Part 2 demonstrated experimentally that the period is dependent on the square root of the length. While the lab procedure was straightforward, the observed high percent error could be attributed to inaccuracies in length and angle measurements, representing inherent challenges in human precision.

Updated: Feb 29, 2024