91Ó°ÊÓ

DATA Experimenting with pendulums, you attach a light string to the ceiling and attach a small metal sphere to the lower end of the string. When you displace the sphere \(2.00 \mathrm{~m}\) to the left, it nearly touches a vertical wall; with the string taut, you release the sphere from rest. The sphere swings back and forth as a simple pendulum, and you measure its period \(T\). You repeat this act for strings of various lengths \(L\), each time starting the motion with the sphere displaced \(2.00 \mathrm{~m}\) to the left of the vertical position of the string. In each case the sphere's radius is very small compared with \(L\). Your results are given in the table: $$ \begin{array}{l|rrrrrrrr} \boldsymbol{L}(\mathbf{m}) & 12.00 & 10.00 & 8.00 & 6.00 & 5.00 & 4.00 & 3.00 & 2.50 & 2.30 \\ \hline \boldsymbol{T}(\mathbf{s}) & 6.96 & 6.36 & 5.70 & 4.95 & 4.54 & 4.08 & 3.60 & 3.35 & 3.27 \end{array} $$ (a) For the five largest values of \(L,\) graph \(T^{2}\) versus \(L\). Explain why the data points fall close to a straight line. Does the slope of this line have the value you expected? (b) Add the remaining data to your graph. Explain why the data start to deviate from the straight-line fit as \(L\) decreases. To see this effect more clearly, plot \(T / T_{0}\) versus \(L,\) where \(T_{0}=2 \pi \sqrt{L / g}\) and \(g=9.80 \mathrm{~m} / \mathrm{s}^{2} .\) (c) Use your graph of \(T / T_{0}\) versus \(L\) to estimate the angular amplitude of the pendulum (in degrees) for which the equation \(T=2 \pi \sqrt{L / g}\) is in error by \(5 \%\).

Step by step solution

01

Create a Graph of Squared Period vs. Length

First plot the squared period \(T^2\) against the pendulum length \(L\) for the five largest values of \(L\) according to the data in the table. The result will be a straight line. The reason is because \(T^2\) is directly proportional to \(L\) in the equation of time period of pendulum \(T=2 \pi \sqrt{L / g}\). When you square both sides, you have \( T^2 = (2 \pi)^2 \times \frac{L}{g}\). Therefore, a plot of \(T^2\) versus \(L\) yields a straight line with slope \((2 \pi)^2/g\). Compare this slope with the expected value which comes from the theoretical equation.

02

Add remaining data and note deviations

Now add the remaining four smaller values of \(L\) to the graph and observe any deviations from the ideal linear relationship. The reason is under the non-ideal conditions (i.e., significant angular amplitudes and presence of air resistance), the period of a pendulum is not necessarily given exactly by \(T=2 \pi \sqrt{L / g}\). The divergence from the linear relationship indicates the presence of these non-ideal factors.

03

Calculate the non-dimensional time period

Next, calculate \(T / T_{0}\) for all the length values, where \(T_{0}=2 \pi \sqrt{L / g}\) and \(g=9.80 \mathrm{~m} / \mathrm{s}^{2}\). Plot this ratio versus \(L\) for a clearer picture of the deviations from the ideal relationship.

04

Estimate the angular amplitude

From the plot of \(T / T_{0}\) versus \(L\), look for the value of \(L\) where the ratio \(T / T_{0}\) begins to show a 5% deviation from 1. This deviation indicates that the simple pendulum principles are starting to break down due to large angular amplitudes. Given the angular amplitude, you can estimate the degree of the amplitude using the small angle approximation.

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Key Concepts

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

Pendulum Period

When studying the period of a pendulum, it's essential to understand that the period, denoted as \(T\), is the time it takes for a pendulum to complete one full oscillation back and forth. For a simple pendulum, under ideal conditions, this period is determined by the formula:\[T = 2\pi \sqrt{\frac{L}{g}}\]where \(L\) is the length of the pendulum and \(g\) is the acceleration due to gravity.

The formula reveals the relationship between period and pendulum length. As the length \(L\) increases, the period \(T\) also increases, as the square root of the length grows. This dependency showcases why even small changes in \(L\) can noticeably alter the pendulum's period, an important factor when considering the setup and measurement of a real-world pendulum experiment.

Graph Analysis

In any scientific experiment, analyzing graphs is a vital step for visualizing relationships between variables. When you graph \(T^2\) versus \(L\) for the largest pendulum lengths, theoretically, the points should align closely to a straight line. This linearity arises from the squared form of the pendulum period equation:
\[T^2 = (2\pi)^2 \frac{L}{g}\]
The slope of this graph should match the value \((2\pi)^2/g\), confirming the mathematical model's accuracy.

As you include data for smaller \(L\) values, you may notice a deviation from the linear fit. This divergence suggests the presence of other variables affecting the experiment, such as increased angular amplitude. Analyzing the graph allows you to see where the assumptions of the simple pendulum model begin to falter.

Non-Ideal Conditions

In real-world pendulum experiments, conditions can often deviate from the ideal. These non-ideal conditions affect the pendulum's behavior, causing deviations from the expected results. Some factors include:

• High Angular Amplitudes: When the pendulum swings with a significant angle, the simple harmonic motion approximation breaks down.
• Air Resistance: The friction caused by air can dampen the pendulum's motion, altering its period.
• String Flexibility: If the string is not ideally rigid, it might bend or stretch, affecting the motion.

These factors lead to a period that does not align perfectly with the theoretical period calculated using \(T = 2\pi \sqrt{L/g}\). Acknowledging these non-ideal conditions is essential when conducting experiments as they will influence the accuracy of your results.

Angular Amplitude

Angular amplitude refers to the maximum angle through which the pendulum swings from its rest position. In the ideal model for a simple pendulum, small angles (typically under 15 degrees) are assumed, which simplifies the equations governing the period.

However, as the angular amplitude increases, the approximation of simple harmonic motion becomes less accurate. This discrepancy manifests as increases or decreases in the actual period compared to the predicted period using \(T = 2\pi \sqrt{L/g}\).

To estimate the angular amplitude where the period differs by 5%, you would examine the graph plotting \(T/T_0\) vs. \(L\). Identifying the length where the plotted line begins deviating significantly from \(T/T_0 = 1\) highlights the breakdown of the small angle assumption.

Physics Experimentation

Conducting a pendulum experiment is not only about understanding the physics behind the motion but also mastering the art of scientific experimentation. Through this process, one learns how to:

• Collect Data Systematically: Ensure each measurement is precise, using consistent methods across multiple trials.
• Analyze and Interpret Data: Use graphs and mathematical models to see patterns and discrepancies.
• Consider Sources of Error: Recognize and mitigate potential errors, like measuring inaccuracies or environmental influences.

The skills developed through experimenting with pendulums extend beyond just pendulum physics. They equip you with a scientific mindset applicable across various fields of study, laying a foundation for complex analysis and critical thinking in future projects.