As you go down into a deep well from the Earth's surface, how does your weight change?

Introduction / Context:
This question tests understanding of how gravitational acceleration varies with depth inside the Earth. Many students know that weight is the force with which the Earth attracts a body, but they often assume it is constant near the surface. In reality, gravitational acceleration changes slightly with height above the surface and with depth below it. The question asks specifically what happens when you go down into a well, which is a conceptual application of the formula for gravity inside the Earth.

Given Data / Assumptions:

• We are considering going down into a well, i.e., moving slightly below the Earth's surface.
• The depth of the well is very small compared to the Earth's radius.
• Weight is defined as W = m * g, where g is the local acceleration due to gravity.
• The Earth is assumed approximately spherical and of uniform density for the basic concept.

Concept / Approach:
Outside or on the surface of the Earth, g is approximately constant. Inside a uniformly dense sphere, gravitational acceleration decreases linearly with distance from the centre. As you move from the surface into the interior, the distance from the centre decreases, and only the mass enclosed within that radius contributes to gravity. Thus, g becomes smaller inside the Earth. Since weight is proportional to g, going down into a well results in a slight decrease in weight, not an increase.

Step-by-Step Solution:
Step 1: Recall that weight W = m * g, where m is constant and g depends on position. Step 2: At the Earth's surface, g has its familiar value of about 9.8 m/s^2. Step 3: Inside the Earth, for a radius r less than the Earth's radius R, the effective mass producing gravity decreases, so g(r) is less than g at the surface. Step 4: When you go down a small depth h in a well, your distance from the centre becomes R - h, and the local g decreases slightly. Step 5: Because W = m * g and g is now slightly smaller, your weight decreases slightly as you go down the well.

Verification / Alternative check:
We can compare this with what happens when you go up a mountain. At higher altitudes, your distance from the Earth's centre increases, and g also decreases slightly, so your weight becomes a bit less. Inside the Earth, a similar reduction in g occurs, but for a different physical reason (less enclosed mass). Both situations lead to a reduction in weight, though the change is very small for moderate heights or depths. This consistency of reasoning supports the conclusion that your weight decreases slightly when you go down into a well.

Why Other Options Are Wrong:
It increases slightly: Incorrect, because moving towards the centre inside the Earth reduces the effective gravitational pull. It remains exactly the same: Not true; while the change is small, it is not zero according to the theory of gravity inside a sphere. None of the above: Incorrect because one of the given options correctly describes the behaviour. It first increases then decreases: This pattern is not valid for small depths in a well; g inside a uniform Earth does not increase below the surface.

Common Pitfalls:
Many students think that moving closer to the Earth's centre must always increase gravity, forgetting that inside a spherical shell the outer layers do not contribute to the gravitational pull on an interior point. Another common misconception is that g is completely constant near the surface, leading to the answer that weight remains exactly the same. Remembering that only the mass enclosed within a given radius contributes to gravity inside the Earth helps avoid such mistakes.

Final Answer:
As you go down into a deep well, your weight decreases slightly because the local gravitational acceleration becomes a little smaller inside the Earth.