Problem 5 - Applied Physics

Question

Find the length (in in.) of a pendulum with a period of \(2.25 \mathrm{~s}\).

Step-by-Step Solution

Answer

The pendulum length is approximately 49.61 inches.

Step 1: Understanding the Problem

To find the length of a pendulum, we need to use the formula that relates the period of the pendulum to its length. The formula is given by the equation for the period of a simple pendulum: \[ T = 2\pi \sqrt{\frac{L}{g}} \]where \( T \) is the period, \( L \) is the length of the pendulum, and \( g \) is the acceleration due to gravity, approximately \( 9.8 \text{ m/s}^2 \).

Step 2: Rearranging the Formula

First, solve the formula for \( L \). Start by isolating \( \sqrt{L} \):\[ \sqrt{\frac{L}{g}} = \frac{T}{2\pi} \]Squaring both sides gives:\[ \frac{L}{g} = \left(\frac{T}{2\pi}\right)^2 \]Multiply both sides by \( g \) to get:\[ L = g \left(\frac{T}{2\pi}\right)^2 \]

Step 3: Substituting Known Values

Now substitute the known values into the equation. We have \( T = 2.25 \text{ s} \) and \( g = 9.8 \text{ m/s}^2 \):\[ L = 9.8 \left(\frac{2.25}{2\pi}\right)^2 \]

Step 4: Calculating the Length

First, calculate the fraction:\[ \frac{2.25}{2\pi} \approx 0.3581 \]Next, square the result:\[ (0.3581)^2 \approx 0.1282 \]Finally, multiply by 9.8 to find \( L \):\[ L = 9.8 \times 0.1282 \approx 1.26 \text{ meters} \]

Step 5: Converting Meters to Inches

Since we need the length in inches, convert \( L = 1.26 \text{ meters} \) to inches. There are approximately 39.37 inches in a meter:\[ 1.26 \text{ meters} \times 39.37 \approx 49.61 \text{ inches} \]

Key Concepts

Simple Pendulum Formula

The simple pendulum formula is a fundamental equation in physics that describes the motion of a pendulum. It relates the period of a pendulum to its length and the gravitational acceleration. The formula is: \[ T = 2\pi \sqrt{\frac{L}{g}} \]where:

• \( T \) is the period of the pendulum (time for one complete cycle).
• \( L \) is the length of the pendulum.
• \( g \) is the acceleration due to gravity, usually approximated as \( 9.8 \text{ m/s}^2 \) on Earth's surface.

The beauty of this formula is in its simplicity. It shows that the period of a pendulum is independent of its mass and depends only on its length and the gravitational field strength. This makes it a great tool for exploring the basics of harmonic motion.

Physics Problem-Solving

When approaching physics problems, it is crucial to follow a structured approach to find a solution. Start by understanding the problem and identifying what is given and what is being asked.For pendulum problems, you'll primarily use the formula for the period of a simple pendulum. Here are the steps to approach these types of problems:

• Identify the given values, such as the period \( T \), and other constants like \( g \).
• Solve for the unknown variable, re-arrange the formula if necessary.
• Substitute the known values and solve step-by-step.
• Make sure the result makes sense with the problem context.

Using these problem-solving strategies can greatly aid in understanding and resolving complex physics problems.

Unit Conversion

In physics, converting units is often necessary to match the required units for a solution or to better interpret results. In this pendulum problem, you calculate the pendulum's length in meters first, and then convert it to inches.The conversion factor from meters to inches is 39.37 inches per meter. Here’s a quick guide to converting:

• Write down the original measurement in meters.
• Multiply by the conversion factor (39.37).
• Round or approximate as needed.

In this example, you found the pendulum's length as approximately 1.26 meters. Conversion gives:\[ 1.26 \text{ meters} \times 39.37 \approx 49.61 \text{ inches} \]This step ensures that your answer is in the desired units as requested in the problem.

Period of Pendulum

The period of a pendulum is the time it takes to complete one full swing, returning to its starting point. Understanding how this is calculated is key in pendulum problems. It depends on two main factors:

• The length of the pendulum \( L \).
• The gravitational acceleration \( g \).

To find the period, the formula used is \( T = 2\pi \sqrt{\frac{L}{g}} \). Rearranging to solve for \( L \) when \( T \) is given involves isolating \( L \) on one side:\[ L = g \left(\frac{T}{2\pi}\right)^2 \]In our example, with a period of 2.25 seconds, you calculated the needed pendulum length by substituting the given values. This calculation reflects how the period grows longer with increasing pendulum length or higher gravity, providing insights into pendulum-based timekeeping and measurements.