Question:
A uniform spherical planet (Radius R) has acceleration due to gravity at its surface g. Points P and Q located inside and outside the planet have acceleration due to gravity g/4 . Maximum possible separation between P and Q is ? Explain in detail.

Answer:

1. Introduction:
We are given a uniform spherical planet with radius R and acceleration due to gravity g at its surface. We need to find the maximum possible separation between two points, P and Q, where the acceleration due to gravity at P is g/4 and at Q is also g/4.

2. Understanding the problem:
To solve this problem, we need to understand the relationship between the acceleration due to gravity and the distance from the center of the planet. On a uniform spherical planet, the acceleration due to gravity decreases as we move away from the center. We also need to consider the fact that the acceleration due to gravity inside the planet is zero.

3. Analyzing the situation:
Let's analyze the situation at points P and Q separately:

a. Point P:
At point P, the acceleration due to gravity is g/4. Since this point is inside the planet, we know that the acceleration due to gravity is zero at the center of the planet. As we move away from the center towards point P, the acceleration due to gravity gradually increases. Therefore, the maximum possible distance of point P from the center of the planet can be determined by equating the acceleration due to gravity at point P with g/4.

b. Point Q:
At point Q, the acceleration due to gravity is also g/4. Since this point is outside the planet, we know that the acceleration due to gravity increases as we move away from the center of the planet. Therefore, the maximum possible distance of point Q from the center of the planet can be determined by equating the acceleration due to gravity at point Q with g/4.

4. Calculating the maximum separation:
To calculate the maximum possible separation between points P and Q, we need to find the maximum possible distance of point P from the center of the planet (denoted as r1) and the maximum possible distance of point Q from the center of the planet (denoted as r2). The maximum separation can be calculated as the sum of r1 and r2.

5. Solution:
Let's calculate the maximum possible distance of point P from the center of the planet (r1) and the maximum possible distance of point Q from the center of the planet (r2):

a. Point P:
Using the formula for the acceleration due to gravity inside a uniform spherical planet, we have:
g/4 = (4/3) * π * G * ρ * r1
where G is the gravitational constant and ρ is the density of the planet. Since the planet is uniform, we can express the density as ρ = M / (4/3 * π * R^3), where M is the mass of the planet. Substituting this expression for ρ into the equation, we have:
g/4 = (4/3) * π * G * (M / (4/3 * π * R^3)) * r1
Simplifying the equation, we get:
g = G