Problem 205 - Calculus Volume 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 calculated value of \( T \) is 0.3\%.

Step 1: Write the Given Formula

The formula for the period of a simple pendulum is given by \( T = 2 \pi \sqrt{\frac{L}{g}} \). Here, \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity.

Step 2: Differentiate the Formula

To find how small changes in \( L \) and \( g \) affect \( T \), we need to differentiate the formula. The differential of \( T \) is given by \( dT = \frac{\partial T}{\partial L} dL + \frac{\partial T}{\partial g} dg \).

Step 3: Calculate Partial Derivatives

Calculate \( \frac{\partial T}{\partial L} \) and \( \frac{\partial T}{\partial g} \):- \( \frac{\partial T}{\partial L} = \pi \left( \frac{1}{\sqrt{Lg}} \right) \)- \( \frac{\partial T}{\partial g} = -\pi \left( \frac{L}{2g^{3/2}} \right) \)

Step 4: Express \( dT \) in Terms of \( dL \) and \( dg \)

Substitute these values back into the differential equation:\[dT = \pi \left( \frac{1}{\sqrt{Lg}} \right) dL - \pi \left( \frac{L}{2g^{3/2}} \right) dg\]

Step 5: Calculate Maximum Percentage Error

Using relative errors, \( \frac{dL}{L} = 0.005 \) and \( \frac{dg}{g} = 0.001 \), compute the maximum percentage error in \( T \):\[\left| \frac{dT}{T} \right| \approx \frac{1}{2} \left( \frac{dL}{L} \right) + \left| \frac{1}{2} \left( \frac{dg}{g} \right) \right|\]Substitute the errors and calculate:\[\left| \frac{dT}{T} \right| \approx \frac{1}{2}(0.005) + \frac{1}{2}(0.001) = 0.003\]

Step 6: Convert to Percentage

The relative error in \( T \) is \( 0.003 \) which is \( 0.3\% \). Thus, the maximum percentage error in the calculated value of \( T \) is \( 0.3\% \).

Key Concepts

Simple Pendulum

A simple pendulum is an idealized model consisting of a mass, known as a bob, attached to a string or rod of fixed length, swinging back and forth under the influence of gravity. These pendulums are commonly used in studying harmonic motion, as they exhibit periodic behavior, meaning they repeat their motion over consistent time intervals. The period, denoted as \( T \), is the time it takes for one complete cycle of the pendulum's motion. For small oscillations, the period of a simple pendulum is given by the formula:\[ T = 2 \pi \sqrt{\frac{L}{g}} \]Where:\- \( L \) is the length of the pendulum.- \( g \) is the acceleration due to gravity.This formula shows that the period of the pendulum is dependent on both the length of the pendulum and the gravitational acceleration.

Partial Derivatives

Partial derivatives are a fundamental concept in calculus, especially when dealing with functions of multiple variables. They represent the rate of change of a function concerning one variable while keeping other variables constant. In the context of the simple pendulum, we want to see how changes in the parameters \( L \) and \( g \) affect the period \( T \). To do this, we take the partial derivatives of \( T \) concerning both \( L \) and \( g \):

• \( \frac{\partial T}{\partial L} = \pi \left( \frac{1}{\sqrt{Lg}} \right) \)
• \( \frac{\partial T}{\partial g} = -\pi \left( \frac{L}{2g^{3/2}} \right) \)

These derivatives help us understand how slight errors in measuring \( L \) and \( g \) will affect the calculated period \( T \). By using partial derivatives, we can separate and analyze the impact of each variable independently.

Percentage Error

Percentage error is a way to quantify the accuracy of a measurement or calculation by comparing the error size to the measured value itself. In the context of the simple pendulum, we calculate the percentage error to determine how the uncertainties in \( L \) and \( g \) translate into uncertainties in the period \( T \). The formula for maximum percentage error in \( T \) is given as:\[ \left| \frac{dT}{T} \right| \approx \frac{1}{2} \left( \frac{dL}{L} \right) + \frac{1}{2} \left( \frac{dg}{g} \right) \]Here, \( \frac{dL}{L} \) and \( \frac{dg}{g} \) represent the relative errors in \( L \) and \( g \) respectively. When we plug in the given errors, \( 0.5\% \) for \( L \) and \( 0.1\% \) for \( g \), we find that the percentage error in \( T \) is \( 0.3\% \). This calculation provides a measure of how sensitive the period is to changes in the pendulum's length and gravitational acceleration.

Calculus

Calculus is the branch of mathematics that deals with continuous change and provides the tools for analyzing dynamic systems. In analyzing the simple pendulum, calculus, particularly the concept of differentials, plays a crucial role.A differential is an infinitesimally small change in a variable, and it helps us understand how changes in one variable affect another. By employing the differential approach, we approximate the impact of small errors in measurements on the period \( T \).In the procedure:

• We find the differential \( dT \) using the partial derivatives of \( T \), then combine them as \( dT = \frac{\partial T}{\partial L} dL + \frac{\partial T}{\partial g} dg \).
• This helps in approximating changes in the outcome, such as calculating percentage errors.

Calculus, thus, allows us to perform nuanced analyses and build models that can predict behavior in real-world systems, even under the influence of small uncertainties.