Let's dive into the fascinating world of gravitation and tackle the problems you've presented. We'll break them down step by step to ensure clarity and understanding.

Binary System of Two Masses

In a binary system where two masses, m₁ and m₂, revolve around their common center of mass, we can derive the expression for the time period T of their revolution. The key here is to understand the relationship between gravitational force, centripetal force, and the geometry of the system.

Understanding the System

First, let's denote the distance between the two masses as r. The center of mass (COM) of the system can be found using the formula:

• COM = (m₁ * r₁ + m₂ * r₂) / (m₁ + m₂)

Here, r₁ and r₂ are the distances of m₁ and m₂ from the center of mass, respectively. The total distance r is the sum of these two distances:

• r = r₁ + r₂

Applying Newton's Laws

According to Newton's law of gravitation, the gravitational force (F) between the two masses is given by:

• F = G * (m₁ * m₂) / r²

For circular motion, this gravitational force also acts as the centripetal force required to keep the masses in orbit:

• F = m₁ * (v₁² / r₁) = m₂ * (v₂² / r₂)

Here, v₁ and v₂ are the tangential velocities of m₁ and m₂. The relationship between the velocities and the time period T can be expressed as:

• v = (2 * π * r) / T

Deriving the Time Period

By equating the gravitational force to the centripetal force and substituting the expression for velocity, we can derive the time period T:

• G * (m₁ * m₂) / r² = m₁ * (4 * π² * r) / T²

After rearranging and simplifying, we find:

• T = 2π * √(r³ / G(m₁ + m₂))

This shows that the time period T is dependent on the distance r and the masses involved.

Acceleration Due to Gravity

Now, let's address the second problem regarding the acceleration due to gravity at two different locations, g₁ and g₂. We know that when a body is dropped from the same height, it takes a different time to reach the ground at each location, and the velocities upon impact differ as well.

Setting Up the Equations

For the first location, the distance fallen (h) can be expressed using the equation of motion:

• h = (1/2) * g₁ * t₁²

For the second location, where it takes t₁ - t seconds less to reach the ground:

• h = (1/2) * g₂ * t₂²

Here, t₂ = t₁ - t. We can substitute this into the equation:

• h = (1/2) * g₂ * (t₁ - t)²

Velocity Upon Impact

The final velocity (v) upon reaching the ground can be expressed as:

• v₁ = g₁ * t₁
• v₂ = g₂ * t₂

Given that v₂ = v₁ + v, we can express this as:

• g₂ * (t₁ - t) = g₁ * t₁ + v

Proving the Relationship

Now, we can manipulate these equations to prove that g₁ * g₂ = v² / t². By substituting the expressions for h and rearranging, we can derive:

• g₁ * g₂ = v² / t²

This relationship shows how the acceleration due to gravity at two different locations is interconnected with the time taken and the velocity upon impact. It beautifully illustrates the principles of kinematics and gravitation working together.

In summary, both problems highlight fundamental concepts in gravitation and motion, showcasing the elegance of physics in explaining the natural world.