College Physics
College Physics
11th Edition
ISBN: 9781305952300
Author: Raymond A. Serway, Chris Vuille
Publisher: Cengage Learning
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# Exploring the Period of a Pendulum

## 1. Graphing the Period as a Function of Length

### a. Drawing the Graph
On the provided axes, draw a graph representing the period of a pendulum (in seconds) as a function of its length (in meters).

### b. Analyzing the Graph's Shape
Explain the shape of the graph you have drawn. Is it linear or curved? Why?

### c. Doubling the Period
If you wanted to double the period of a pendulum, what change would you make to its length? Explain using two points on the line you have drawn \((L_1, T_1)\) and \((L_2, T_2)\) where \(T_2\) is \(2 \times T_1\), and the required change in length is accurately represented.

### Figure 11-2:
A graph is provided with the x-axis labeled "LENGTH (m)" and the y-axis labeled "PERIOD (s)." 

---

## Explanation of the Graph
The period of a pendulum \(T\) is related to its length \(L\) by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]

From this relationship, we can observe the following:

- When \(L\) is plotted on the x-axis and \(T\) on the y-axis, the graph will be a curve, not a straight line. This is because \(T\) increases as the square root of \(L\), leading to a parabolic shape when plotted.
- To double the period \(T\) of a pendulum, you would need to increase the length \(L\) by a factor of four. This is derived from the relationship \(T \propto \sqrt{L}\). So, if \(T_2 = 2 \times T_1\), then \((L_2 = 4 \times L_1)\).
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Transcribed Image Text:# Exploring the Period of a Pendulum ## 1. Graphing the Period as a Function of Length ### a. Drawing the Graph On the provided axes, draw a graph representing the period of a pendulum (in seconds) as a function of its length (in meters). ### b. Analyzing the Graph's Shape Explain the shape of the graph you have drawn. Is it linear or curved? Why? ### c. Doubling the Period If you wanted to double the period of a pendulum, what change would you make to its length? Explain using two points on the line you have drawn \((L_1, T_1)\) and \((L_2, T_2)\) where \(T_2\) is \(2 \times T_1\), and the required change in length is accurately represented. ### Figure 11-2: A graph is provided with the x-axis labeled "LENGTH (m)" and the y-axis labeled "PERIOD (s)." --- ## Explanation of the Graph The period of a pendulum \(T\) is related to its length \(L\) by the formula: \[ T = 2\pi \sqrt{\frac{L}{g}} \] From this relationship, we can observe the following: - When \(L\) is plotted on the x-axis and \(T\) on the y-axis, the graph will be a curve, not a straight line. This is because \(T\) increases as the square root of \(L\), leading to a parabolic shape when plotted. - To double the period \(T\) of a pendulum, you would need to increase the length \(L\) by a factor of four. This is derived from the relationship \(T \propto \sqrt{L}\). So, if \(T_2 = 2 \times T_1\), then \((L_2 = 4 \times L_1)\).
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