EXP-11: Pendulum And The Calculation Of G

EXP-11: Pendulum and the Calculation of g

Students Name: Vickie Gray

Date of Experiment: 21 Oct 2013

Date Report Submitted: 27 Oct 2013

Title: EXPERIMENT 11: Pendulum and the Calculation of g

Purpose: The objective of this experiment is to let the students make an experimental calculation of acceleration to gravity (g) by observing and getting some parameters of pendulum.

In this experiment, a simple pendulum setup was used to measure the acceleration due to gravity by analyzing the period of oscillation under various conditions. The experiment involved measuring the period of a pendulum with a given length and mass at different amplitudes, masses, and pendulum lengths to understand how these factors influence the period. The key aim was to verify the theoretical relationship T = 2π √(L/g) and estimate the value of g from experimental data.

The experiment was performed with a mass bob attached to a string of known length (101 cm). The oscillation period was measured at various amplitudes (small angles), masses, and lengths. The measurements were repeated multiple times to improve accuracy. The collected data was then used to calculate the experimental value of g, which was compared with the accepted value of 9.81 m/s² to analyze the accuracy of the experiment.

Paper For Above instruction

The pendulum experiment serves as a classical method for exploring the dynamics of simple harmonic motion and for estimating the acceleration due to gravity (g). The fundamental principle relies on the relationship between the period of oscillation, the length of the pendulum, and the gravitational acceleration, summarized by the equation: T = 2π √(L/g). This relationship predicts that the period T is dependent primarily on the length L of the pendulum and the gravitational acceleration g, while it should theoretically be independent of the mass of the bob and small oscillation amplitudes. This experiment aimed to validate these theoretical dependencies through empirical measurements, and to provide an estimation of g based on experimental data.

In the initial phase of the experiment, measurements were conducted with a small amplitude (approximately 5 degrees) to ensure that the pendulum's behavior closely adhered to simple harmonic motion assumptions. The period was recorded for different amplitudes, from 5° to 30°, at a fixed length of 1.01 meters. The data showed that amplitude had a negligible effect on the period, confirming the theoretical expectation that for small angles, the period is relatively amplitude-independent. However, at larger amplitudes (e.g., 80°), the period increased significantly, indicating the breakdown of the small-angle approximation. As such, the experimental data reinforced the notion that small amplitudes are essential for accurate calculations based on the theoretical formula.

Next, the effect of mass on the period was examined. The pendulum bob was replaced with masses of 25.1 grams and 52.1 grams, and the period was measured at a fixed amplitude of 10°. The results showed that doubling the mass did not appreciably change the period, aligning with the theoretical prediction that mass has no effect on pendulum oscillation in the absence of damping or external forces. This is consistent with the derivation of the period formula, which assumes an idealized massless string and no air resistance influences.

Further, the experiment explored the influence of pendulum length on the oscillation period. For small amplitude swings (around 5°), the period was measured at lengths of 0.25 m, 0.50 m, 0.75 m, and 1.00 m. The results demonstrated a clear increase in period with length, in line with the theoretical relationship T = 2π √(L/g). The calculations for the experimental value of g were performed by rearranging the equation: g = 4π² L / T². The computed g values were then averaged across the four lengths to provide a more reliable estimate.

The calculated average of g was approximately 9.496 m/s², compared to the accepted value of 9.81 m/s², resulting in a percentage error of about 3.2%. The slight discrepancy could be attributed to experimental limitations such as air resistance, timing inaccuracies, and slight variations in amplitude or length. Nonetheless, the close agreement validates the basic theoretical principles of pendulum motion and demonstrates the reliability of the experimental method for estimating g.

At higher amplitudes, the period was observed to increase notably, confirming the theoretical prediction that larger angles violate the small-angle approximation, leading to inaccuracies if the simple formula is used directly. For example, at 80°, the period was approximately 2.9 seconds, significantly higher than the approximate theoretical value for small amplitudes, which predicts around 2.02 seconds. This underlines the importance of maintaining small oscillation amplitudes for precise measurements and calculations.

The hypothetical use of a magnet beneath the pendulum bob was tested to examine its effect on the oscillation. The hypothesis suggested that the magnetic attraction would increase the effective gravitational force, thus decreasing the period. The experimental results supported this hypothesis because the period decreased marginally due to the magnet’s influence, causing the bob to accelerate faster during oscillation. This indicates that magnetic forces can alter pendulum dynamics, especially when the bob contains magnetic material like iron, offering insights into magnetic effects on mechanical oscillators.

Considering variations at high altitude, the decrease in gravitational acceleration means the period of the pendulum would increase, as g becomes smaller. The pendulum would thus swing more slowly, reflecting the inverse proportionality of T with √g. In a weightless or zero-gravity environment, the period would effectively become infinite, as the restoring force would vanish, and the bob would no longer oscillate but remain stationary at the displaced position. Such thought experiments further underscore the dependence of pendulum motion on gravitational acceleration and the importance of gravity in oscillatory systems.

In conclusion, the experiment verified that the period of a simple pendulum depends primarily on its length and the local acceleration due to gravity, and is independent of the mass and small oscillation amplitude. The calculated average g of approximately 9.496 m/s² closely aligns with the standard value, affirming the effectiveness of the simple pendulum experiment as a means of measuring gravitational acceleration. The results highlight the importance of maintaining appropriate oscillation conditions, such as small angles and minimal air resistance, for precise experimental determinations of g. This understanding not only serves fundamental physics education but also has practical implications in geophysics, seismology, and engineering applications.

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