Problem 25 - Master Resource Book in JEE Main Physics

Question

There is a mine of depth about \(2.0 \mathrm{~km}\). In this mine the conditions as compared to those at the surface are (a) lower air pressure, higher acceleration due to gravity (b) higher air pressure, lower acceleration due to gravity (c) higher air pressure, higher acceleration due to gravity (d) lower air pressure, lower acceleration due to gravity

Step-by-Step Solution

Answer

(b) Higher air pressure, lower acceleration due to gravity.

Step 1: Understanding the Factors Affecting Air Pressure

Air pressure is the force exerted by the weight of air above a surface. At a greater depth compared to the Earth's surface, such as in a mine 2 km deep, the air pressure is generally higher because the column of air above is taller and thus heavier. Hence, deeper locations tend to have higher air pressure.

Step 2: Analyzing Gravity Within Earth

Gravity is affected by the distance from the Earth's center. As you move towards the center of the Earth, the acceleration due to gravity slightly decreases because less of the Earth's mass is beneath you. At a 2 km depth, the reduction in gravity is small but existent, resulting in slightly lower gravity compared to at sea level.

Step 3: Evaluating Each Option

Let's evaluate the reasons behind each choice. Option (a) suggests lower air pressure, but it is incorrect because air pressure increases with depth. Option (c) suggests higher air pressure and higher gravity, but gravity decreases slightly with depth. Option (d) indicates both air pressure and gravity are lower, which doesn't match the higher air pressure fact. Thus, the correct answer is (b) which states higher air pressure and lower gravity at greater depths.

Key Concepts

Air Pressure

Air pressure is an essential concept that describes the force exerted by air molecules on any surface in contact with them. Imagine air as a collection of countless tiny particles bouncing around in the atmosphere. Each tiny molecule collides with surfaces, creating pressure.

When we look into how air pressure changes with depth, such as in a mine 2 km beneath the Earth's surface, it’s intriguing. As you go deeper, the column of air above you increases. More air molecules mean more weight pushing down, causing the air pressure to rise.

In other words:

• At higher altitudes, air pressure is lower because there's less air above exerting force downward.
• Conversely, at lower altitudes or greater depths like in a mine, air pressure is higher due to more air being stacked overhead.

Hence, when you descend into a mine, even as shallow as 2 km, you experience significantly higher air pressure compared to the surface. This detail is crucial to understanding how air behaves in different layers of the Earth's atmosphere.

Depth of Mine

The concept of 'depth of mine' revolves around how different environmental factors change as we move further beneath the Earth's surface. The mine's depth is an important variable that influences both air pressure and gravity.

Going deeper affects various physical conditions:

• Air Pressure: As mentioned earlier, deeper locations have a taller column of air above them, leading to increased air pressure.
• Acceleration Due to Gravity: Gravity slightly decreases as depth increases due to the changing distribution of Earth's mass.

At a depth of 2 km, these changes might be subtle, but they are noticeable enough to affect readings. It's fascinating to note how even small shifts in environment, like descending into a mine, can demonstrate natural Earth's laws impacting physical properties.

Earth's Center

The journey towards the Earth's center is a journey through physics. Earth’s gravity is a perplexing yet fascinating aspect of our planet. A common notion is that gravity decreases as you get closer to Earth's core. This is due to the gravity's reliance on the mass beneath you.

In simple terms:

• Gravity depends on how much of Earth's mass is below you. More mass means a stronger gravitational pull.
• As you move deeper beneath the Earth's surface, less mass is beneath you, thus slightly reducing the gravity experienced.

This concept helps explain why, in a mine, the acceleration due to gravity is marginally lower than at the surface.

However, the difference at a depth of just 2 km is not huge. It's an amazing demonstration of how gravity is influenced by Earth's layers and distribution of mass, impacting how we measure and experience forces in underground environments.