Problem 13 - Applied Physics

Question

What is the period of a pendulum that is \(0.25 \mathrm{~m}\) long?

Step-by-Step Solution

Answer

The period of the pendulum is approximately 1.002 seconds.

Step 1: Understanding the Formula

The period of a pendulum is given by the formula: \( T = 2\pi \sqrt{\frac{L}{g}} \), where \( T \) is the period, \( L \) is the length of the pendulum, and \( g \) is the acceleration due to gravity (approximately \( 9.81 \mathrm{~m/s^2} \) on Earth).

Step 2: Substitute Values

Plug the given length of the pendulum \( L = 0.25 \mathrm{~m} \) and the acceleration due to gravity \( g = 9.81 \mathrm{~m/s^2} \) into the formula: \( T = 2\pi \sqrt{\frac{0.25}{9.81}} \).

Step 3: Calculate Inner Square Root

First, compute the fraction under the square root: \( \frac{0.25}{9.81} \approx 0.02548 \). Then calculate the square root: \( \sqrt{0.02548} \approx 0.1596 \).

Step 4: Calculate the Period

Now, calculate the period by multiplying the square root by \( 2\pi \): \( T = 2\pi \times 0.1596 \approx 1.002 \). The period of the pendulum is approximately \( 1.002 \) seconds.

Key Concepts

Physics Formula

In physics, formulas serve as a fundamental method of describing the laws and behaviors of nature. When it comes to a pendulum, a particular formula helps us determine the time it takes for one complete cycle or swing, known as the period. This is given by the formula: \( T = 2\pi \sqrt{\frac{L}{g}} \). Here:

• \( T \) stands for the period, which measures the time for one oscillation.
• \( L \) represents the pendulum's length, a crucial factor in determining the period.
• \( g \) is the gravitational acceleration, which on Earth is roughly \( 9.81 \, \mathrm{m/s^2} \).

Understanding this formula is key to mastering concepts involving pendulum motion.

Gravitational Acceleration

Gravitational acceleration is the force that the Earth exerts to attract objects towards its center. This constant is fundamental in the physics of pendulum motion. On Earth, its average value is about \( 9.81 \, \mathrm{m/s^2} \).

• This constant helps determine how fast a pendulum swings back and forth.
• The formula for a pendulum's period uses gravitational acceleration to calculate the speed and timing of the pendulum's swing.
• Though it varies slightly depending on location on Earth's surface, \( 9.81 \, \mathrm{m/s^2} \) is typically used as a standard value for calculations.

Gravitational acceleration makes sure that the pendulum doesn't just swing aimlessly, but does so in a predictable and timed manner.

Simple Harmonic Motion

A pendulum exhibits a classic example of simple harmonic motion (SHM), which occurs when an object moves back and forth through an equilibrium position. This movement can be described as follows:

• The pendulum, when released, swings back and forth in a regular, repeating path that is symmetric around a central point.
• The forces that act upon it try to return it to its equilibrium position, making it a restoring force motion.
• Because of these forces, once disturbed, the pendulum continuously rocks back and forth, creating what we refer to as periodic motion.

This regularity of motion is why pendulums are often used in clocks—they tick away at consistent intervals due to their simple harmonic characteristics.

Pendulum Length

The length of a pendulum, denoted as \( L \), is one of the key factors that influence its period. This is how it relates to pendulum behavior:

• Longer pendulums have longer periods, meaning they swing more slowly.
• Shorter pendulums swing faster, resulting in shorter periods.
• The relationship is not linear; the period is proportional to the square root of the length, as indicated in the formula \( T = 2\pi \sqrt{\frac{L}{g}} \).

This means doubling the length of a pendulum doesn’t double its period; instead, it increases by a factor of the square root of 2, showing the delicate balance in pendulum dynamics.