Problem 41 - College Physics

Question

A science museum has asked you to design a simple pendulum that will make 25.0 complete swings in \(85.0 \mathrm{~s}\). What length should you specify for this pendulum?

Step-by-Step Solution

Answer

The pendulum should be approximately 2.87 meters long.

Step 1: Understand the Pendulum Period Formula

A simple pendulum's period (the time for one complete swing back and forth) is determined by the formula: \( T = 2\pi\sqrt{\frac{L}{g}} \), where \( T \) is the period in seconds, \( L \) is the length of the pendulum in meters, and \( g \) is the acceleration due to gravity (approximately \( 9.81 \text{ m/s}^2 \) on Earth).

Step 2: Calculate the Period of One Swing

To find the period of one swing, divide the total time given by the number of swings: \( T = \frac{85.0 \text{ s}}{25.0} \). This gives us \( T = 3.4 \text{ s} \).

Step 3: Rearrange the Formula to Solve for Length

Using the formula for the period \( T = 2\pi\sqrt{\frac{L}{g}} \), rearrange it to solve for \( L \): \( L = \left(\frac{T}{2\pi}\right)^2 g \).

Step 4: Plug the Values into the Formula

Substitute \( T = 3.4 \text{ s} \) and \( g = 9.81 \text{ m/s}^2 \) into the rearranged formula: \( L = \left(\frac{3.4}{2\pi}\right)^2 \times 9.81 \).

Step 5: Calculate the Length

Calculate the length using the values: \( L = \left(\frac{3.4}{6.2832}\right)^2 \times 9.81 \approx 2.87 \text{ m} \).

Key Concepts

Pendulum Period

A crucial aspect of understanding simple pendulums is the concept of the period. The period of a pendulum refers to the time it takes for the pendulum to make one full back-and-forth swing. For a simple pendulum, which assumes the pendulum's length l is much greater than the size of the oscillating object, the period is calculated using the formula: \[ T = 2\pi\sqrt{\frac{L}{g}} \]where:

• \(T\) is the period of the pendulum (in seconds),
• \(L\) is the length of the pendulum (in meters),
• \(g\) is the acceleration due to gravity (standard is approximately \(9.81 \text{ m/s}^2\) on Earth).

This relationship highlights how the period is influenced predominately by the length of the pendulum and the gravitational force acting on it. Notably, the period is independent of the mass of the pendulum or the amplitude of its swing, provided the angles are small.

Gravity

Gravity plays a pivotal role in the functioning of pendulums. As a force that pulls objects towards the center of the Earth, it is what causes the pendulum to swing back and forth. In the pendulum period formula, \(g\) represents the acceleration due to Earth's gravity, which is roughly \(9.81 \text{ m/s}^2\). This constant is vital for calculating the period of a pendulum and helps determine how quickly or slowly it will swing. In different environments—like the Moon or Mars—where gravity is weaker, the pendulum would have a longer period, meaning it swings slower. On Earth, however, this consistent gravitational force makes pendulum calculations precise and reliable for various applications ranging from clocks to educational experiments.

Physics Problem Solving

Solving physics problems, such as those involving pendulums, requires a structured approach. Begin by understanding the problem's requirements and identifying the given variables. This involves interpreting the scenario correctly, such as determining the total time a pendulum swings and the number of complete swings to calculate the pendulum's period. Next, apply relevant formulas. In the case of the pendulum problem, use the period formula: \[ T = 2\pi\sqrt{\frac{L}{g}} \]Then isolate the variable you need to solve for, in this case, the length \(L\). Rearrange the formula accordingly:\[ L = \left(\frac{T}{2\pi}\right)^2 \times g \]Insert the known values, such as \(T\) and \(g\), into this rearranged equation to find the unknown length. This systematic approach helps ensure accuracy and efficiency in problem-solving, reducing errors often encountered in physics calculations.

Pendulum Length Calculation

Calculating the length of a pendulum involves using the rearranged pendulum period formula. Once the period \(T\) is known, you can determine the length \(L\) necessary for a pendulum to achieve a specific period using:\[ L = \left(\frac{T}{2\pi}\right)^2 \times g \]For our museum pendulum, we first calculated the period of one swing:

• Given total time: 85.0 s,
• Number of swings: 25.0,
• Period \(T = \frac{85.0}{25.0} = 3.4 \text{ s}\).

Using \(T = 3.4 \text{ s}\) and \(g = 9.81 \text{ m/s}^2\), plug them into the formula:\[ L = \left(\frac{3.4}{6.2832}\right)^2 \times 9.81 \approx 2.87 \text{ m} \]This ensures that the designed pendulum will meet the specification of 25 swings in 85 seconds, providing a practical and dependable way to design pendulum-based systems.