Problem 53 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 53-56: Solve for the specified variable. $$ T=2 \pi \sqrt{\frac{l}{g}} \text { for } l \quad \text { (period of a pendulum) } $$

Step-by-Step Solution

Answer

The solution for \( l \) is \( l = \frac{g \, T^2}{(2\pi)^2} \).

Step 1: Isolate the Square Root

To solve for \( l \), we first need to isolate the square root on the right-hand side of the equation. Our equation is \( T = 2\pi \sqrt{\frac{l}{g}} \). Divide both sides by \(2\pi\): \[ \frac{T}{2\pi} = \sqrt{\frac{l}{g}} \].

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation: \[ \left( \frac{T}{2\pi} \right)^2 = \left( \sqrt{\frac{l}{g}} \right)^2 \]. This simplifies to: \[ \frac{T^2}{(2\pi)^2} = \frac{l}{g} \].

Step 3: Solve for the Variable

To solve for \( l \), multiply both sides by \( g \): \[ l = g \cdot \frac{T^2}{(2\pi)^2} \]. This gives us the expression for \( l \).

Key Concepts

Solving Equations

When faced with an equation, your main goal is to find the unknown variable. This process is known as solving equations. Equations are like puzzles where you need to determine what value makes the statement true. They often contain mathematical expressions set equal to one another, and your task is to figure out the missing piece, which in most cases is a variable like \( l \) in the pendulum formula.To solve an equation:

• Read the equation carefully to understand which variable you need to solve for.
• Apply algebraic techniques to systematically isolate the desired variable.
• Always check your results by substituting the variable back into the original equation.

In our original exercise, we start with the equation for the period of a pendulum. To solve for \( l \), you need to manipulate the equation step by step until \( l \) is isolated on one side of the equation.

Algebraic Manipulation

Algebraic manipulation is all about rearranging terms and applying mathematical operations to an equation. It involves strategies like adding, subtracting, multiplying, or dividing both sides of the equation by the same number without changing its equality.Key Techniques:

• Dividing both sides of an equation evenly, as seen when we divide by \( 2\pi \) to start isolating the square root.
• Squaring both sides to remove a square root, a crucial step to simplify our equation further in terms of \( l \).

The goal of algebraic manipulation is to clarify the equation and bring the variable you are interested in into focus. In our pendulum problem, we apply algebraic manipulation to transition from the original equation form to the one where \( l \) is explicitly solved.

Isolation of Variables

Isolation of variables means separating the variable you want to solve for from other parts of the equation. It is often the final step after the initial manipulation.The ultimate aim here is to rewrite the equation so that the variable stands alone on one side of the equation. This often involves:

• Canceling out terms not involving the variable. For instance, we multiplied by \( g \) to isolate \( l \) in the final step.
• Using operations that simplify to remove extraneous parts of the expression.

In our pendulum equation, this process started with isolating the square root and squaring both sides of the equation, and it concluded by multiplying to clear the division by \( g \). This leads us directly to the formula for \( l \) in terms of \( T \), \( g \), and constants.