Problem 24 - 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) In order to double the period, how would we have to change the length \(l\)?

Step-by-Step Solution

Answer

(a) \( T = k \sqrt{l} \). (b) Quadruple the length to double the period.

Step 1: Understanding the relationship

The problem states that the period of a pendulum varies directly with the square root of its length. This is a direct variation, which can be expressed mathematically as \( T \propto \sqrt{l} \), where \( T \) is the period, and \( l \) is the length of the pendulum. To write this as an equation, introduce a constant of proportionality, \( k \), resulting in the equation \( T = k \sqrt{l} \).

Step 2: Expressing the equation (part a)

Using the direct variation relationship from the previous step, the equation representing the period \( T \) in terms of the pendulum's length \( l \) is \( T = k \sqrt{l} \), where \( k \) is a proportionality constant.

Step 3: Analyzing how to double the period (part b)

To determine how to double the period \( T \), we set the new period to \( 2T \) and substitute into the equation: \( 2T = k \sqrt{l'} \), where \( l' \) is the new length. Substituting \( T = k \sqrt{l} \) from the original equation, \( 2(k \sqrt{l}) = k \sqrt{l'} \). Solving for \( l' \): \( 2\sqrt{l} = \sqrt{l'} \). Squaring both sides, we get \( 4l = l' \).

Step 4: Conclusion: Change in length

To double the period of the pendulum, the length \( l \) must be quadrupled, given by the equation \( l' = 4l \).

Key Concepts

Direct Variation

In mathematics, direct variation refers to a situation where two quantities change in the same ratio. This means if one quantity increases, the other increases as well, and if one decreases, the other does too. Direct variation can be expressed mathematically with the equation \( y = kx \), where \( y \) and \( x \) are the quantities in question, and \( k \) is the constant of proportionality. In our pendulum example, the period \( T \) of the pendulum directly varies with the square root of its length \( l \). Here, \( k \) is a specific constant that maintains the proportionality between \( T \) and \( \sqrt{l} \).
To put it simply, whenever you see a direct variation, you can think of two things moving in harmony, always kept in balance by a constant factor. This knowledge is crucial for understanding how changes in length will affect the pendulum's swing period.

Proportional Relationship

In everyday language, a proportional relationship is when two values maintain a consistent ratio as they change. This concept is like having a steady rhythm or synchrony between two participants in a dance.
Mathematically, a proportional relationship can be captured by the equation \( T = k \sqrt{l} \), which relates the period \( T \) of a pendulum to its length \( l \).
When exploring the topic of pendulums, this proportionality is crucial for predicting how the pendulum behaves when its length changes. For instance, if the length of a pendulum is quadrupled, its swing period will be doubled due to this specific mathematical relationship. It's through understanding this proportional link that we can anticipate and explain how and why certain changes in one quantity lead to specific changes in another.

Square Root

The square root is a mathematical function that finds a number which, when multiplied by itself, gives the original number. It's denoted by the symbol \( \sqrt{\cdot} \).
In the context of pendulums, understanding the square root is key to comprehending how the pendulum's period changes with its length. The relationship \( T = k \sqrt{l} \) tells us that the period \( T \) increases as the square root of the length \( l \) increases.
To further illustrate, if you want to double the period of the pendulum, you'll notice that the square root implies a nonlinear change in length. Specifically, you will need to make the pendulum's length four times longer, because \( (\sqrt{4} = 2) \) results in doubling the square root value. This demonstrates how a change in the length yields a different rate of change in the period when dealing with square roots, highlighting their unique impact in proportional relationships.