Problem 34 - Elementary and Intermediate Algebra

Question

Pendulums. The time it takes for one complete swing of a pendulum varies directly with the square root of the length \(L\) of the pendulum. Write an equation that models this type of variation. CAN'T COPY THE IMAGE

Step-by-Step Solution

Answer

The equation is \(T = k \cdot \sqrt{L}\).

Step 1: Understanding Direct Variation

In this scenario, we're told that the period of a pendulum's swing, denoted by time \(T\), varies directly with the square root of the length \(L\) of the pendulum. A direct variation means \(T\) is proportional to \(\sqrt{L}\). This can be expressed as \(T = k \cdot \sqrt{L}\), where \(k\) is a constant of proportionality.

Step 2: Formulate the Equation

Given that \(T\) varies directly with \(\sqrt{L}\), we can set up the equation for this relationship as \(T = k \cdot \sqrt{L}\). Here, \(k\) is a constant that would be determined based on further information, such as specific measurements or conditions of the pendulum's swing.

Key Concepts

Pendulum Period

The period of a pendulum refers to the time it takes to complete one full swing, from one side to the other and back again. Pendulums are fascinating due to their rhythmic and predictable motion. This makes them useful in measuring time, as seen in traditional clocks.
The period depends on the length of the pendulum, where the longer the pendulum, the longer the period. It's crucial to understand that factors such as the mass of the pendulum bob or the amplitude of the swing have negligible effects on the period, as long as the swings remain small. For small angles, the period ( T ) is directly tied to the length ( L ), paving the way for a mathematical relationship.

Square Root Relationship

In the study of pendulums, the period (T) shows a direct relationship with the square root of the pendulum's length (L). This means that as the length increases, the period increases in such a manner that \( T \) gets multiplied by the square root of the change in length. For example:

• Doubling the length of a pendulum increases the period by a factor of \( \sqrt{2} \).
• Similarly, tripling the length results in a period increase by \( \sqrt{3} \).

This relationship is beautifully captured in the formula \( T = k \cdot \sqrt{L} \), where \( k \) is a constant. It's fascinating because it highlights how a relatively simple change in length has a predictable effect on the period, an essential principle for designing time-measuring devices.

Proportionality Constant

The proportionality constant, denoted as \( k \), is crucial in the equation \( T = k \cdot \sqrt{L} \). This constant holds the key to converting the relationship into a precise numerical prediction. It depends on factors that are intrinsic to the pendulum, such as the strength of gravity and the specific settings of the experiment.
Determining \( k \) typically requires measurements under controlled conditions. Once \( k \) is known, it allows us to calculate the pendulum's period for any given length with accuracy. Understanding \( k \) means grasping how consistent and predictable pendulum motion can be, which is exactly why pendulums have been pivotal in timekeeping and scientific investigations.