Question

write an equation that relates the quantities. The period of a pendulum is the time required for one oscillation; the pendulum is usually referred to as simple when the angle made to the vertical is less than \(5^{\circ} .\) The period \(T\) of a simple pendulum (in seconds) varies directly with the square root of its length \(l\) (in feet). The constant of proportionality is \(\frac{2 \pi}{\sqrt{32}}\)

Short answer

The equation relating the period \( T \) and the length \( l \) is \( T = \frac{\pi}{2 \sqrt{2}} \sqrt{l} \).

Step-by-step solution

Understand the Problem

The problem states that the period of a simple pendulum varies directly with the square root of its length. The period is denoted as \( T \) and the length as \( l \). Additionally, we are given the constant of proportionality \( k \).

Identify the Direct Variation

A direct variation relationship between the period \( T \) and the square root of the length \( l \) can be written as: \[ T = k \, \times \, \sqrt{l} \]

Substitute the Constant of Proportionality

We know that the constant of proportionality \( k \) is given as \( \frac{2 \pi}{\sqrt{32}} \). Substitute this value into the equation: \[ T = \left( \frac{2 \pi}{\sqrt{32}} \right) \, \times \, \sqrt{l} \]

Simplify the Equation

Simplify the equation to express the final relationship. Since \( \frac{2 \pi}{\sqrt{32}} = \frac{2 \pi}{4 \sqrt{2}} = \frac{\pi}{2 \sqrt{2}} \), the equation is: \[ T = \frac{\pi}{2 \sqrt{2}} \sqrt{l} \]

Key concepts

direct variation

Direct variation is a mathematical concept that describes a direct relationship between two variables. If one variable increases, the other variable increases proportionally, and vice versa. In a direct variation, one can be written as a constant multiple of the other. For example, if you see a formula like:

• T varies directly as the square root of l

It means there is a direct variation between T and the square root of l. Therefore, we can write: \( T = k \sqrt{l} \)where k is the constant of proportionality. In our case, T is the period of the pendulum, and l is the length of the pendulum. Direct variation is significant because it simplifies how we describe relationships between variables. Once you know the constant, it is easy to compute one variable if you know the other. You will see this form often in physics and other sciences.

simple pendulum

A simple pendulum is a weight suspended from a pivot, so it can swing freely. When displaced from its equilibrium position, it is subjected to a restoring force due to gravity, which will oscillate it back-and-forth. Simple pendulums have some key characteristics:

• They assume a small displacement angle (usually less than 5 degrees) from the vertical.
• Their motion can be approximated as simple harmonic motion.

This means that the period of the oscillation (the time it takes for one complete cycle) depends on the length of the string and the acceleration due to gravity but not on the amplitude of the swing. For small angles, this is an excellent approximation and is useful for computations in physics and engineering.

proportionality constant

In mathematics and physics, a proportionality constant is a value that relates two variables that are directly proportional to each other. In our problem, the relationship between the period of the pendulum (T) and the square root of its length (\( \sqrt{l} \)) is expressed using a proportionality constant k. This appears in the formula: \( T = k \sqrt{l} \)In this specific case, the proportionality constant is given as \( \frac{2 \, \pi}{\sqrt{32}} \). This constant combines various factors like the gravitational constant that affect the motion of the pendulum. By substituting \( k \) into the direct variation equation, you can see how the period relates to the length of the pendulum. Proportionality constants make it easier to handle direct variations by providing a straightforward way to compute one variable when the other is known.

square root relationship

A square root relationship occurs when one variable is proportional to the square root of another. This is often seen in various physical phenomena, including the period of a pendulum. In this context, it means that: \( T \propto \sqrt{l} \)or \( T = k \sqrt{l} \)where T is the period of the pendulum and l is its length. The square root relationship implies that if you quadruple the length of the pendulum, the period will only double. This is because the period grows with the square root of the length.This relationship is crucial as it simplifies the understanding of how changes in length affect the period. It tells us that to double the period of a pendulum, you need to increase the length by a factor of four, not two. Such understanding aids in designing experiments and interpreting the results in physics and engineering.