Problem 30 - Algebra and Trigonometry

Question

Law of the Pendulum The period of a pendulum (the time elapsed during one complete swing of the pendulum) varies directly with the square root of the length of the pendulum. (a) Express this relationship by writing an equation. (b) To double the period, how would we have to change the length \(R\).

Step-by-Step Solution

Answer

The length must be quadrupled to double the period.

Step 1: Understand Pendulum Period Relation

The period of a pendulum, denoted by \( T \), varies directly with the square root of its length \( L \). This can be expressed in the form \( T = k \sqrt{L} \), where \( k \) is a proportionality constant.

Step 2: Express the Relation as an Equation

Since the period \( T \) is directly proportional to the square root of the length \( L \), we write the equation \( T = k \sqrt{L} \).

Step 3: Determine the Alteration for Doubling the Period

To double the period \( T \), we need \( 2T = k \sqrt{R} \). Since the original \( T = k \sqrt{L} \), we substitute and solve: \( 2k \sqrt{L} = k \sqrt{R} \). This simplifies to \( \sqrt{R} = 2 \sqrt{L} \).

Step 4: Solve for the New Length

Square both sides of \( \sqrt{R} = 2 \sqrt{L} \) to eliminate the square root. We get \( R = 4L \). This means we need to quadruple the length of the pendulum to double the period.

Key Concepts

Direct Variation

Direct variation is a mathematical relationship where one variable changes in direct proportion to another. In simpler terms, when one variable increases, the other one does too, and vice versa. This kind of relationship is common in everyday phenomena and is often expressed using the formula:

• \( y = kx \)

Here, \( y \) and \( x \) are the variables, while \( k \) stands for the proportionality constant that makes the equation hold true. In our pendulum example, the period \( T \) of a pendulum varies directly with the square root of its length \( L \). Therefore, the formula becomes \( T = k \sqrt{L} \). This equation shows that if you increase the square root of the length \( L \), the period \( T \) will proportionally increase by the same factor when \( k \) remains consistent.
Recognizing direct variation is crucial in solving problems related to physical phenomena, such as understanding how changing one aspect of an experiment affects another. This insight helps you know how to form and manipulate the equations to find desired results.

Proportionality Constant

The proportionality constant, commonly denoted as \( k \), is a crucial element in direct variation equations that scales the relationship between two variables. It essentially represents the "rate" at which one variable affects another, and it can differ based on the context or scenario you're dealing with.
In the context of the pendulum, the equation \( T = k \sqrt{L} \) uses the proportionality constant to link the pendulum's period \( T \) with the square root of its length \( L \). While \( k \) itself remains unspecified unless measurements are taken, it is constant for a particular setting, such as a specific pendulum under uniform conditions. This means that under identical conditions, a pendulum's changes are predictable and consistent due to this constant factor.
Understanding \( k \) can also aid in real-world applications, like engineering or physics, where precise calculations are essential. The constant helps define how significantly one factor, such as pendulum length, influences another, like the period, which is crucial for designing systems that rely on pendulum motion.

Square Root Function

The square root function is a mathematical operation that finds a number which, when multiplied by itself, gives the original number. It is often represented as \( \sqrt{x} \) or \( x^{1/2} \) and shows up frequently in various equations. In the context of pendulums, it plays a central role in determining pendulum period because the period \( T \) depends on the square root of the length \( L \), written as \( T = k \sqrt{L} \).
To understand the behavior of a system governed by a square root, consider that this function grows slower than a linear one. Doubling \( L \) does not double \( \sqrt{L} \), but instead an increase of \( L \) results in a lesser relative increase in \( T \). This property explains why, when you need to double the period, you have to quadruple the length.
Square root functions have wide-ranging applications, not limited to pendulum problems. They are vital in fields like statistics, physics, and engineering, enabling calculations that illustrate relationships between different physical quantities. Understanding how they work and interact with other mathematical functions can deepen your insight into natural processes and technological designs.