A pendulum swings with a period \( T \) given by \( T = 2\pi \sqrt{\frac{L}{g}} \), where \( g = 9.8 \, \text{m/s}^2 \) and \( T = 4 \, \text{s} \). Solve for the length \( L \).

["Solving for Pendulum Length: How to Calculate the Swing Period", "The pendulum’s motion follows a precise scientific law that connects its period ( T ), length ( L ), and gravitational acceleration ( g ):\n[ T = 2\pi \sqrt{\frac{L}{g}} ]", "Understanding this formula empowers you to determine the length of a pendulum given its swing period—essential in physics, engineering, and timekeeping. In this article, we’ll solve for the pendulum length ( L ) using real-world values: ( T = 4 , \ ext{seconds} ) and ( g = 9.8 , \ ext{m/s}^2 ).", "### Understanding the Period Equation", "The period ( T ) is the time it takes for a pendulum to complete one full swing back and forth. The equation:\n[ T = 2\pi \sqrt{\frac{L}{g}} ]\nrelates this observable time period to the physical length ( L ) and gravity ( g ), which remains nearly constant on Earth.", "### Step-by-Step Solution", "Start with the formula:\n[\nT = 2\pi \sqrt{\frac{L}{g}}\n]", "1. Isolate the square root term:\nDivide both sides by ( 2\pi ):\n[\n\sqrt{\frac{L}{g}} = \frac{T}{2\pi}\n]", "2. Square both sides to eliminate the square root:\n[\n\frac{L}{g} = \left( \frac{T}{2\pi} \right)^2\n]", "3. Solve for ( L ): Multiply both sides by ( g ):\n[\nL = g \left( \frac{T}{2\pi} \right)^2\n]", "### Plug in Known Values", "Given:\n- ( T = 4 , \ ext{s} )\n- ( g = 9.8 , \ ext{m/s}^2 )", "Substitute:\n[\nL = 9.8 \left( \frac{4}{2\pi} \right)^2 = 9.8 \left( \frac{2}{\pi} \right)^2\n]", "Compute:\n[\n\left( \frac{2}{\pi} \right)^2 = \frac{4}{\pi^2} \approx \frac{4}{9.8696} \approx 0.4053\n]\n[\nL \approx 9.8 \ imes 0.4053 \approx 3.973 , \ ext{meters}\n]", "### Final Answer", "Rounded to two decimal places, the length ( L ) of the pendulum is:\n[\n\boxed{L \approx 3.97 , \ ext{m}}\n]", "This means a pendulum swinging with a period of 4 seconds must be approximately 3.97 meters long under standard gravity. This formula provides a clear quantitative link between physical length and timing, fundamental to both educational demonstrations and practical applications like clock design.", "---", "Ready to calculate other pendulum lengths? Use the formula ( L = g \left( \frac{T}{2\pi} \right)^2 ) and a calculator to explore different periods and gravitational conditions."]