Problem 3 - Applied Physics

Question

Find the period of a pendulum \(1.25 \mathrm{~m}\) long.

Step-by-Step Solution

Answer

The period of the pendulum is approximately 2.243 seconds.

Step 1: Understand the Pendulum Formula

The period of a pendulum is determined by the length of the string and the acceleration due to gravity. The formula for the period \( T \) of a simple pendulum is given by: \[ T = 2\pi \sqrt{\frac{L}{g}} \]where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity, approximately \(9.81 \text{ m/s}^2\).

Step 2: Substitute the Given Length

In this step, we'll substitute the given length of the pendulum \( L = 1.25 \text{ m} \) into the formula: \[ T = 2\pi \sqrt{\frac{1.25}{9.81}} \]

Step 3: Calculate the Square Root

Calculate the square root of the fraction \( \frac{1.25}{9.81} \):\[ \sqrt{\frac{1.25}{9.81}} \approx \sqrt{0.1274} \approx 0.357 \]

Step 4: Compute the Period

Finally, we compute the period by multiplying \(2\pi\) with the square root value we found:\[ T = 2\pi \times 0.357 \approx 2.243 \]

Step 5: Conclusion: Determine the Period

The period of the pendulum, calculated using the given length, is approximately \(2.243\) seconds.

Key Concepts

Simple Pendulum

A simple pendulum is a fascinating physical concept that involves a point mass, referred to as the "bob," suspended from a fixed point by a weightless, inextensible string. When displaced from its equilibrium position and released, it oscillates back and forth in a regular periodic motion. The simple pendulum is an idealization, assuming no air resistance and a perfect point mass at the end.

Here are the key characteristics of a simple pendulum:

• It consists of a mass attached to a string or rod of fixed length, which doesn't stretch.
• The only forces acting on it are gravitational force and tension in the string.
• The pendulum exhibits harmonic motion, which means it swings back and forth in a regular cycle.

The motion of a simple pendulum is important in demonstrating the principles of harmonic motion and energy conservation. Understanding how it behaves helps in explaining more intricate concepts in physics, such as oscillations, resonance, and dynamics.

Pendulum Formula

The pendulum formula is vital to determine how long it takes for a pendulum to complete one full swing, known as the period. The period (T) is the time it takes for the pendulum to return to its original position and is dependent on a few specific factors.

This formula is expressed as:\[T = 2\pi \sqrt{\frac{L}{g}}\]Where:

• T is the period of the pendulum.
• L is the length of the pendulum string.
• g is the acceleration due to gravity, typically approximated as 9.81 m/s² on the surface of the Earth.

This relationship shows that the period increases with the length of the pendulum and is unaffected by the mass. The formula is derived under the assumption of small angle approximations, meaning that the swing angles are comparatively small. Understanding this equation allows you to predict how changes in pendulum length or gravitational influence affect the period of oscillation.

Acceleration Due to Gravity

Acceleration due to gravity ( g ) is a fundamental constant in the pendulum period calculation. It represents how fast an object accelerates when in free fall near the Earth's surface. Typically, this value is taken as approximately 9.81 m/s². However, it can vary slightly depending on altitude and geographical location.

Here are some important aspects about it:

• It is the force that pulls objects toward the Earth, causing them to accelerate downward.
• In pendulum calculations, this constant allows for understanding how fast the pendulum swings back and forth.
• Variations in g slightly influence pendulum motion; a stronger gravitational field results in a faster period.

The significance of g in pendulum-related concepts extends beyond simple physics problems. It's crucial in designing timekeeping devices like pendulum clocks, where precise measurements of g ensure accurate time.