Accuracy in the Determination of Gravitational Acceleration (g) Using a Simple Pendulum

Understanding the accuracy in calculating gravitational acceleration, g, is crucial in physics and engineering. This article will guide you through the process, employing a simple pendulum and basic mathematical techniques.

Introduction to Simple Pendulum and Gravitational Acceleration

A simple pendulum consists of a mass suspended from a fixed point so that it can swing freely. The period of oscillation, T, is given by the formula:

Formula for the Period of a Pendulum

T 2π √(L/g)

From this, we can express gravitational acceleration g as:

Expression for Gravitational Acceleration

g 4π2L/T2

Determining Gravitational Acceleration with Given Data

The length L of the pendulum is 20 cm (0.2 m) with a measurement uncertainty of 1 mm (0.001 m). The time for 100 oscillations is 90 seconds, determined using a wristwatch with a resolution of 1 second. Thus, the period T is:

T 90 s / 100 0.9 s

The uncertainty in the period T, ΔT, can be calculated using:

ΔT 1 s / 100 0.01 s

Now, substituting the values into the formula for g:

g ≈ 4π2(0.2) / (0.9)2 ≈ 9.74 m/s2

Calculating the Uncertainty in Gravitational Acceleration

To determine the uncertainty in g, we use the propagation of uncertainties. The fractional uncertainty in g can be calculated as:

left(frac{Δg}{g}right)^2 left(frac{ΔL}{L}right)^2 2 left(frac{ΔT}{T}right)^2

Substituting the values obtained:

fractional uncertainty in L: ΔL/L 0.001 / 0.2 0.005

fractional uncertainty in T: ΔT/T 0.01 / 0.9 ≈ 0.0111

Total fractional uncertainty in g:

left(frac{Δg}{g}right)^2 ≈ (0.005)^2 2 (0.0111)^2 ≈ 0.000025 0.000246 ≈ 0.000271

Taking the square root:

frac{Δg}{g} ≈ sqrt{0.000271} ≈ 0.0165 or 1.65%

Converting back to the absolute uncertainty:

Δg ≈ g times 0.0165 ≈ 9.74 times 0.0165 ≈ 0.1614 m/s2

Thus, the value of g and its uncertainty is:

g ≈ 9.74 ± 0.16 m/s2

Relative Error in the Determination of g

The relative error in the determination of g can also be determined as:

Δg/g ΔL/L 2 ΔT/T

Substituting the values:

Δg/g 0.005 2 * 0.0001 0.0052 or 0.52%

Conclusion

In this article, we have explored how to accurately determine gravitational acceleration using a simple pendulum. The detailed calculations and propagation of uncertainties have provided a comprehensive understanding of the accuracy of the determination.

Keywords: Pendulum period, gravitational acceleration, uncertainty propagation