Problem 59 - Calculus with Concepts in Calculus

Question

The period \(T\) of a pendulum of length \(L\) that swings under the influence only of gravity is given approximately by \(T=2 \pi \sqrt{L / g}\), where \(g=9.8\) meters per second squared. a. Write \(L\) as a function of \(T\). b. The length of the Foucault pendulum at the Smithsonian Institution is approximately \(21.8\) meters. Determine its period.

Step-by-Step Solution

Answer

a. \( L(T) = g \left(\frac{T}{2\pi}\right)^2 \) b. The period is approximately \( 9.37 \) seconds.

Step 1: Rearrange the Period Formula

The formula for the period of a pendulum is given by \( T = 2\pi \sqrt{\frac{L}{g}} \). To express \( L \) as a function of \( T \), we need to solve for \( L \). First, divide both sides by \( 2\pi \) to isolate the square root: \( \frac{T}{2\pi} = \sqrt{\frac{L}{g}} \).

Step 2: Eliminate the Square Root

Square both sides of the equation to remove the square root: \( \left(\frac{T}{2\pi}\right)^2 = \frac{L}{g} \).

Step 3: Solve for L

Multiply both sides by \( g \) to solve for \( L \): \( L = g \left(\frac{T}{2\pi}\right)^2 \). Thus, \( L \) as a function of \( T \) is \( L(T) = g \left(\frac{T}{2\pi}\right)^2 \).

Step 4: Solve for Period Using Given Length

We need to find the period \( T \) when \( L = 21.8 \) meters. Use the period formula: \( T = 2\pi \sqrt{\frac{L}{g}} \). Substitute \( L = 21.8 \) meters and \( g = 9.8 \text{ m/s}^2 \): \( T = 2\pi \sqrt{\frac{21.8}{9.8}} \).

Step 5: Calculate the Period

First, calculate the expression inside the square root: \( \frac{21.8}{9.8} \approx 2.2245 \). Then take the square root: \( \sqrt{2.2245} \approx 1.4915 \). Multiply by \( 2\pi \): \( 2\pi \times 1.4915 \approx 9.37 \). Therefore, the period is approximately \( 9.37 \) seconds.

Key Concepts

Solving equations

Solving equations is all about manipulating them to find the value of an unknown variable. If we have an equation, like the pendulum period formula, we follow a series of steps to isolate the desired variable on one side. In our problem, the equation \( T = 2\pi \sqrt{\frac{L}{g}} \) needs to be solved for \( L \) in terms of \( T \). This involves rearranging the equation step by step.
To start, you'd divide both sides by \( 2\pi \), giving you \( \frac{T}{2\pi} = \sqrt{\frac{L}{g}} \). This step isolates the square root term. Next, you must eliminate the square root to solve for \( L \). You achieve this by squaring both sides, resulting in \( \left(\frac{T}{2\pi}\right)^2 = \frac{L}{g} \).
Finally, solve for \( L \) by multiplying both sides by \( g \), leading to \( L = g \left(\frac{T}{2\pi}\right)^2 \). Now, \( L \) is expressed directly as a function of \( T \). Solving equations often uses such systematic steps, allowing us to tackle even complex symbols and expressions.

Physical constants

Physical constants play a crucial role in equations linking multiple physical quantities. In the pendulum formula \( T = 2\pi \sqrt{\frac{L}{g}} \), \( g \) represents the acceleration due to gravity, a physical constant with a value of approximately 9.8 meters per second squared on the surface of the Earth. This constant characterizes the downward acceleration imparted by gravity on freely falling objects.
Using constants like \( g \) makes calculations reliable because they provide a universal reference point for comparing and predicting results. In any pendulum equation,\( g \) helps in understanding how fast an object will swing under the sole influence of gravity, without any external force.
Thus, physical constants are fundamental to applying laws of physics across various scenarios, ensuring consistency in scientific predictions and equations.

Rearranging formulas

Rearranging formulas is essential to express quantities differently, often needed in solving physics problems. Here, the pendulum period formula \( T = 2\pi \sqrt{\frac{L}{g}} \) needs rearrangement to solve for different variables. This flexibility is valuable when the problem gives quantities we must use differently from how the formula is originally presented.
To rearrange, we follow these principles:

• Isolate the term containing the variable of interest (e.g., \( L \)).
• Use algebraic operations such as division, multiplication, and squaring to further manipulate the formula.
• Maintain equality by applying the same operation to both sides of the formula.

Using these rules, you ensure the integrity of the equation while adapting it to suit different contexts.
Rearranging formulas is therefore not only a technical skill but also a conceptual tool to illuminate relationships between different academic subjects, like mathematics and physics.