Problem 205 - Calculus 3

Question

The period \(T\) of a simple pendulum with small oscillations is calculated from the formula \(T=2 \pi \sqrt{\frac{L}{g}},\) where \(L\) is the length of the pendulum and \(g\) is the acceleration resulting from gravity. Suppose that \(L\) and \(g\) have errors of, at most, 0.5\(\%\) and \(0.1 \%,\) respectively. Use differentials to approximate the maximum percentage error in the calculated value of \(T .\)

Step-by-Step Solution

Answer

The maximum percentage error in the period \(T\) is approximately 0.25\%.

Step 1: Understand the question

We need to find the maximum percentage error in the period \(T\) of a simple pendulum using differentials. We are given the formula \(T=2\pi\sqrt{\frac{L}{g}}\), and the errors in \(L\) and \(g\) are at most 0.5\(\%\) and 0.1\(\%\), respectively.

Step 2: Differentiate the formula for T

We start by differentiating \(T=2\pi\sqrt{\frac{L}{g}}\) with respect to \(L\) and \(g\). Using the chain rule, the differential of \(T\) is given by: \[dT = \frac{\partial T}{\partial L}dL + \frac{\partial T}{\partial g}dg.\] Calculating the partial derivatives:\[\frac{\partial T}{\partial L} = \pi\sqrt{\frac{1}{Lg}}, \]\[\frac{\partial T}{\partial g} = -\pi\sqrt{\frac{L}{g^3}}.\]

Step 3: Calculate differential changes

Substitute the partial derivatives into the expression for \(dT\):\[dT = \pi\sqrt{\frac{1}{Lg}}dL - \pi\sqrt{\frac{L}{g^3}}dg.\]

Step 4: Relate differentials to percentage errors

Since percentage error is calculated as \(\frac{dQ}{Q} \times 100\%\), we express \(dL\) and \(dg\) in terms of their respective percentage errors:\[\frac{dL}{L} = 0.5\%, \quad \frac{dg}{g} = 0.1\%.\]

Step 5: Substitute errors into differential expression

Substitute the errors into the expression for the differential of \(dT\):\[\frac{dT}{T} = \frac{1}{2}\left(\frac{dL}{L} + \frac{g}{3}\frac{dg}{g}\right) = \frac{1}{2}(0.5\%) + \frac{1}{2}(-0.1\%) = 0.25\%.\]

Step 6: Calculate maximum percentage error

The maximum percentage error in \(T\) is obtained by adding the absolute values of the component percentage errors, thus:\(\Delta T_{\text{max}} = 0.25\%\).

Key Concepts

Simple Pendulum

A simple pendulum is one of the most fundamental mechanical systems used to illustrate classic principles of motion. It consists of a weight, or bob, attached to the end of a string or rod of length \(L\). This setup swings back and forth like a grandfather clock's pendulum. The primary forces acting on the pendulum are gravity and tension from the string or rod.

One of the fascinating properties of a simple pendulum is its periodic motion, which is the repetitive change in position over equal time intervals. The formula used to calculate the period \(T\) of such a pendulum is given by \(T = 2\pi\sqrt{\frac{L}{g}}\), where \(g\) represents the acceleration due to gravity. This formula applies when the pendulum is allowed to make very small angle swings, ensuring that the motion remains simple harmonic.

Key factors influencing the period of a pendulum include:

• Length (\(L\)): Increases in the pendulum's length lead to longer periods.
• Gravity (\(g\)): Stronger gravitational forces decrease the period.

Understanding how to calculate the changes or errors in these factors can be crucial when high precision is required, such as in measuring time or scientific experiments.

Percentage Error

Percentage error is a way to express the accuracy of a measurement by comparing the predicted and actual values. It gives an indication of how far off a measurement is from the real or accepted value. This concept is crucial in science and engineering, where precise measurements make all the difference.

To find the percentage error, you typically use the formula:\[\text{Percentage Error} = \left( \frac{\text{Estimated Value} - \text{Actual Value}}{\text{Actual Value}} \right) \times 100\%\]In the context of our simple pendulum exercise, we were given maximum allowable percentage errors for the variables of \(L\) and \(g\). These were 0.5\% and 0.1\% respectively. Calculating the resulting error in the calculated pendulum period \(T\) is crucial, as even small variations can significantly impact results when high precision is needed.

Here’s why percentage error matters:

• Precision: Identifying potential errors helps refine calculations and predictions.
• Preventing Miscalculations: Helps identify significant sources of error, allowing for corrective measures.

Being able to see where errors can occur, and being aware of their magnitude, is a critical part of scientific accuracy.

Partial Derivatives

Partial derivatives are a fundamental tool in calculus, specifically useful for dealing with functions of more than one variable. They help determine how a change in one variable impacts the function output while keeping other variables constant.

In our exercise, the formula for the period of the simple pendulum involves two variables: \(L\) and \(g\). To approximate errors in \(T\), partial derivatives with respect to each variable were calculated:

• \(\frac{\partial T}{\partial L} = \pi\sqrt{\frac{1}{Lg}}\): Shows how changes in the pendulum's length affect the period.
• \(\frac{\partial T}{\partial g} = -\pi\sqrt{\frac{L}{g^3}}\): Indicates how changes in gravitational acceleration impact the period.

These derivatives were used in the error calculation formula: \(dT = \frac{\partial T}{\partial L}dL + \frac{\partial T}{\partial g} dg\). By understanding each variable’s contribution to the overall error, scientists and engineers can make more accurate predictions and correct for any inaccuracies in measurement or calculation.

Partial derivatives thereby enable the evaluation of each variable separately, offering insights into their individual effects. This is essential in scenarios where multiple factors influence an outcome.