91Ó°ÊÓ

The swinging pendulum Mrs. Hanrahan's precalculus class collected data on the length (in centimeters) of a pendulum and the time (in seconds) the pendulum took to complete one back-and forth swing (called its period). Here are their data: $$ \begin{array}{cc} \hline \text { Length (cm) } & \text { Period (s) } \\ 16.5 & 0.777 \\ 17.5 & 0.839 \\ 19.5 & 0.912 \\ 22.5 & 0.878 \\ 28.5 & 1.004 \\ 31.5 & 1.087 \\ 34.5 & 1.129 \\ 37.5 & 1.111 \\ 43.5 & 1.290 \\ 46.5 & 1.371 \\ 106.5 & 2.115 \\ \hline \end{array} $$ (a) Make a reasonably accurate scatterplot of the data by hand, using length as the explanatory variable. Describe what you see. (b) The theoretical relationship between a pendulum's length and its period is $$ \text { period }=\frac{2 \pi}{\sqrt{g}} \sqrt{\text { length }} $$ where \(g\) is a constant representing the acceleration due to gravity (in this case, \(g=980 \mathrm{~cm} / \mathrm{s}^{2}\) ). Use the following graph to identify the transformation that was used to linearize the curved pattern in part (a). (c) Use the following graph to identify the transformation that was used to linearize the curved pattern in part (a).

Step by step solution

01

Create a Scatterplot

Plot the given data points on a graph such that the length (in cm) is on the x-axis and the period (in seconds) is on the y-axis. This scatterplot will allow us to visually assess the relationship between the length of the pendulum and its period.

02

Analyze the Scatterplot

Upon plotting the data, observe the trend. The points will likely depict a curved, upward-sloping pattern since as the length increases, the period tends also to increase, but not linearly. This suggests a non-linear relationship.

03

Theoretical Relationship

The theoretical relationship between a pendulum's length and its period can be expressed with the equation \( \text{period} = \frac{2\pi}{\sqrt{g}} \sqrt{\text{length}} \). This implies the period is proportional to the square root of the length.

04

Linearization via Transformation

Given the theoretical formula, consider transforming the length data by taking the square root of each length value. Plot the square root of the length against the period. This should linearize the data, as the theoretical model shows the period is proportional to the square root of the length.

05

Identify the Transformation

From the graph mentioned in part (c), identify the transformation used for linearization. In this case, the transformation used is plotting the square root of the length, which linearizes the initially non-linear relationship seen in the scatterplot of length vs. period.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatterplot Analysis

A scatterplot is a powerful tool that visually represents the relationship between two variables. In the context of pendulum experiments, scatterplots help us understand how the length of a pendulum affects its period. This is done by plotting length on the x-axis and period on the y-axis. Creating a scatterplot involves simply marking the data points on a graph based on these two variables.

By connecting the dots they've plotted, students can observe the overall trend or pattern of the data. This is particularly important for identifying the type of relationship between the variables, such as whether they move together in a linear fashion or in a more complex way.

• Scatterplots help identify patterns.
• They can reveal trends or non-trends quickly.

The data from the pendulum exercise reveals a curved, upward-sloping trend. This curvilinear pattern arises because the period increases with pendulum length, but not at a constant rate. Observing this curve is crucial for determining the nature of the relationship and deciding if transformations are needed for better analysis.

Non-linear Relationships

In many real-world situations, including pendulum experiments, relationships between variables are not purely linear. A non-linear relationship means that the rate of change between variables is not constant. This can make data analysis slightly more complicated but also more interesting.

In the case of pendulum length and period, the relationship is non-linear, as represented by the theoretical equation: \[ \text{period} = \frac{2\pi}{\sqrt{g}} \sqrt{\text{length}} \]This formula indicates that the period is proportional to the square root of the pendulum's length. Therefore, as length increases, the period increases as well, but not in a straightforward linear manner.

Such non-linear relationships often require a transformation of data to facilitate better analysis. Recognizing a non-linear relationship is key to choosing the correct mathematical tools or transformations to change the data into a linear form. Understanding this concept plays a vital role in analyzing data that appears curved or follows a pattern difficult to model with a straight line.

Data Transformation

Data transformation is a mathematical process used to change the scale or form of data, primarily to simplify analysis or fulfill assumptions of statistical models.

In experiments like pendulum analysis, transformation is used to turn non-linear data into a linear form. This is a common technique to make relationships more apparent and easier to interpret. In our pendulum exercise, taking the square root of the length is the chosen transformation to linearize the relationship with the period.

• Transformation can make complex data interpretable.
• It helps in meeting model assumptions more readily.

By applying \[ \text{new length} = \sqrt{\text{length}} \]we can plot this transformed length against the period to achieve a straight line. This simplifies recognizing the relationship and fitting a model to the data. Applying the right transformation thus enables us to see the underlying trend more distinctly. Understanding data transformation is fundamental in statistics, particularly when dealing with the diverse and complex forms of data in various fields.