91Ó°ÊÓ

Acceleration due to gravity (\( g \)) is a fundamental concept that refers to the force pulling objects towards the center of a massive body, like Earth. On the Earth's surface, this acceleration is about \( 9.8 \text{ m/s}^2 \). However, this value isn't constant everywhere. It's influenced by factors like altitude and depth.
When you venture inside the Earth, such as into a mine, gravity decreases. This change is calculated using the formula:

• \( g' = g \left(1 - \frac{d}{R}\right) \)

Here, \( d \) is the depth below the Earth's surface, and \( R \) is the Earth's radius. This equation helps determine how gravity weakens as you move down. In our example, at a depth of \( 1600 \text{ km} \), gravity decreases to \( 7.35 \text{ m/s}^2 \). Understanding these changes is crucial for calculating the pendulum's behavior in such environments.

The pendulum length \( L \) is a pivotal parameter in determining the behavior of a simple pendulum. It is the distance from the pivot point to the center of the pendulum bob. The length influences the pendulum's time period, which is the time taken to complete one full oscillation.
In pendulum experiments, it is vital to convert measurements into the same units. For instance, if the pendulum length is given in centimeters, as in our problem, convert it to meters. Thus, \( 40 \text{ cm} \) becomes \( 0.4 \text{ m} \). Using consistent units helps ensure the accuracy of computations.

The time period of a pendulum is calculated using the formula: \( T = 2\pi\sqrt{\frac{L}{g}} \). This expression demonstrates how the time period is affected by the pendulum's length (\( L \)) and the local acceleration due to gravity (\( g \)).
In our scenario, inside a mine with a reduced gravity of \( 7.35 \text{ m/s}^2 \) and a pendulum length of \( 0.4 \text{ m} \), the calculation becomes:

• \( T = 2\pi\sqrt{\frac{0.4}{7.35}} \)
• \( \approx 2\pi\sqrt{0.0544} \)
• \( \approx 2\pi \times 0.233 \approx 1.46 \text{ seconds} \)

This means that it takes approximately \( 1.46 \text{ seconds} \) for the pendulum to complete one oscillation in the mine, which is longer than on the Earth's surface due to the decreased gravity.

The relationship between depth and gravity is an intriguing aspect of physics. As one moves deeper into the Earth, the gravitational force decreases. This phenomenon is due to the shrinking distance to the Earth's center and the distribution of Earth's mass above the object.
The formula \( g' = g \left(1 - \frac{d}{R}\right) \) captures this relationship by showing how gravity (\( g' \)) changes with depth (\( d \)). In simple terms:

• Deeper means less gravity.
• The formula shows that gravity decreases proportionally with depth.
• At significant depths, such as \( 1600 \text{ km} \), the reduction is substantial, with gravity reduced by about 25% from its surface value.

Understanding this relationship is essential for interpreting how pendulums and other phenomena behave under different conditions inside the Earth.