The Problem

Two bodies A and B, with masses 2 kg and 4 kg respectively, are separated by a distance of 2 m. We need to determine the position where a body of mass 1 kg should be placed so that the gravitational force on this body due to bodies A and B is zero.

The Solution

In order to find the position where the gravitational force on the 1 kg body is zero, we can use the concept of gravitational force and the principle of superposition.

Principle of Superposition
According to the principle of superposition, the net gravitational force on an object due to multiple sources is the vector sum of the individual gravitational forces exerted by each source.

Calculating the Gravitational Force
The gravitational force between two bodies can be calculated using Newton's law of universal gravitation, which states that the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

The formula for calculating the gravitational force (F) between two bodies of masses (m1 and m2) separated by a distance (r) is:

F = G * (m1 * m2) / r^2

where G is the gravitational constant (approximately 6.674 × 10^-11 N*m^2/kg^2).

Analysis
In this problem, we have three bodies: A, B, and the 1 kg body. Let's assume that the 1 kg body is placed at a distance x from body A. Therefore, the distance between body B and the 1 kg body would be (2 - x).

Now, we need to calculate the gravitational forces exerted by bodies A and B on the 1 kg body and find the position where the net gravitational force is zero.

Let's calculate the gravitational forces:

- Gravitational force exerted by body A on the 1 kg body:
F1 = G * (m1 * m) / x^2

- Gravitational force exerted by body B on the 1 kg body:
F2 = G * (m2 * m) / (2 - x)^2

According to the problem statement, the net gravitational force on the 1 kg body should be zero. Therefore, the sum of the forces F1 and F2 should be zero.

F1 + F2 = 0

Substituting the equations for F1 and F2, we get:

G * (m1 * m) / x^2 + G * (m2 * m) / (2 - x)^2 = 0

Simplifying the equation:

(m1 * m) / x^2 = - (m2 * m) / (2 - x)^2

Taking the ratio of the masses:

(m1 / m2) = - (x^2 / (2 - x)^2)

Solution
To find the position x where the gravitational force on the 1 kg body is zero, we can solve the above equation for x. However, this equation is non-linear and solving it analytically may not be straightforward.

To find the approximate solution, we can use numerical methods such as iteration or trial and error. We can start by assuming a value for x and calculate the left-hand side of the equation. If

The body should be kept 0.5m near the body A then it will be 0