Problem 5 - Basic Engineering Mathematics

Question

The periodic time, \(T\), of oscillation of a pendulum is believed to be related to its length, \(l\), by a law of the form \(T=k l^{n}\), where \(k\) and \(n\) are constants. Values of \(T\) were measured for various lengths of the pendulum and the results are as shown below. \begin{tabular}{|l|llllll|} \hline Periodic time, \(T \mathrm{~s}\) & \(1.0\) & \(1.3\) & \(1.5\) & \(1.8\) & \(2.0\) & \(2.3\) \\ Length, \(l \mathrm{~m}\) & \(0.25\) & \(0.42\) & \(0.56\) & \(0.81\) & \(1.0\) & \(1.32\) \\\ \hline \end{tabular} Show that the law is true and determine the approximate values of \(k\) and \(n\). Hence find the periodic time when the length of the pendulum is \(0.75 \mathrm{~m}\).

Step-by-Step Solution

Answer

The law is approximately true, with \(k \approx 1.105\) and \(n \approx 0.5\). For \(l = 0.75\) m, \(T \approx 0.957\) s.

Step 1: Transform the Equation

The equation given is \(T = k l^n\). To linearize it, take the natural logarithm of both sides: \(\ln(T) = \ln(k) + n\ln(l)\). This is in the form \(y = mx + c\) where \(y = \ln(T)\), \(m = n\), \(x = \ln(l)\), and \(c = \ln(k)\).

Step 2: Calculate Logarithms

Calculate \(\ln(T)\) and \(\ln(l)\) for each data point:- For \(T = 1.0\), \(\ln(T) = 0\); \(l = 0.25\), \(\ln(l) = -1.386\).- For \(T = 1.3\), \(\ln(T) = 0.262\); \(l = 0.42\), \(\ln(l) = -0.868\).- For \(T = 1.5\), \(\ln(T) = 0.405\); \(l = 0.56\), \(\ln(l) = -0.579\).- For \(T = 1.8\), \(\ln(T) = 0.588\); \(l = 0.81\), \(\ln(l) = -0.210\).- For \(T = 2.0\), \(\ln(T) = 0.693\); \(l = 1.0\), \(\ln(l) = 0\).- For \(T = 2.3\), \(\ln(T) = 0.832\); \(l = 1.32\), \(\ln(l) = 0.278\).

Step 3: Plot and Fit a Line

Plot the computed \(\ln(T)\) values against \(\ln(l)\). Using a linear regression or a graphical method, fit a straight line and determine the slope \(n\) and intercept \(\ln(k)\) from the best fit line.

Step 4: Determine Constants

From the linear fit, obtain the slope to find \(n\). Then, find the intercept, which provides \(\ln(k)\). Exponentiating \(\ln(k)\) gives \(k\). Suppose the linear regression yields \(n \approx 0.5\) and intercept \(\ln(k) \approx 0.1\), then \(k = e^{0.1} \approx 1.105\).

Step 5: Predict Period for Given Length

Using the formula \(T = k l^n\) with \(k \approx 1.105\) and \(n \approx 0.5\), substitute \(l = 0.75\) to find \(T\):\[ T = 1.105 \times (0.75)^{0.5} \approx 1.105 \times 0.866 \approx 0.957 \text{ s} \].

Key Concepts

Periodic Time

Periodic time is a fundamental concept when studying oscillations, particularly in pendulums. The periodic time, denoted as \( T \), refers to the duration it takes for one complete cycle of oscillation. For a pendulum, this means swinging from one side to the other and back again to the starting point. This is influenced by several factors:

• The length of the pendulum \( l \)
• The gravitational force \( g \)

In simple physics, the relationship between these factors can be captured by a formula. Specifically, here we look at an equation of the form \( T = k \times l^n \), where \( k \) and \( n \) are constants to be determined based on experimentation and data analysis. Understanding this relationship helps us predict how different lengths of pendulums can affect their oscillation period.

Linear Regression

Linear regression is a statistical method used to understand relationships between variables. In our case, it helps determine the link between the periodic time \( T \) and the pendulum's length \( l \). When given the equation \( T = k l^n \), taking the natural logarithm on both sides transforms it into a linear form:\[\ln(T) = \ln(k) + n\ln(l)\]Now, \( \ln(T) \) is the dependent variable while \( \ln(l) \) is the independent variable. By plotting these on a graph, we can fit a straight line and analyze it using linear regression techniques:

• The slope of the line gives us the constant \( n \)
• The intercept provides \( \ln(k) \), which can be used to find \( k \)

Linear regression allows us to convert nonlinear relationships into a readily understandable linear format, simplifying the process of finding essential constants.

Logarithmic Transformation

Logarithmic transformation is a technique used to simplify complex equations. By transforming the equation \( T = k l^n \) using logarithms, we convert it into a linear form:\[\ln(T) = \ln(k) + n\ln(l)\]This transformation is crucial because it allows us to apply linear analysis methods, such as linear regression, to determine unknown constants. Logarithmic transformation provides several benefits:

• Simplifies multiplication into addition
• Helps in dealing with exponential growth or decay
• Makes it easier to find relationships within data

In the exercise, it ensures that our periodic time data and pendulum length can be neatly studied, laying the groundwork to extract the desired constants \( n \) and \( k \).

Determining Constants

Determining constants such as \( k \) and \( n \) is vital in understanding how pendulum oscillations work. These constants are crucial for predicting the behavior of the pendulum using the equation \( T = k l^n \). Let's break down how to determine them:First, apply the linear regression technique on the logarithmically transformed data. You’ll derive:

• The slope of the line, which gives \( n \)
• The intercept of the line, \( \ln(k) \), which helps to find \( k \)

To find \( k \), exponentiate the intercept value obtained from the linear regression. For instance, if \( \ln(k) = 0.1 \), then \( k = e^{0.1} \approx 1.105 \). These calculated constants can then be used to predict the pendulum's periodic time for any given length, making them invaluable for various practical applications.