Gravitational Force at Half the Radius of the Earth

To determine the gravitational force on a body at a height equal to half the radius of the Earth, we need to consider the relationship between mass, distance, and gravitational force.

1. Gravitational Force Formula
The gravitational force between two bodies can be calculated using Newton's law of universal gravitation:

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

Where:
- F is the gravitational force between the two bodies
- G is the gravitational constant (approximately 6.67430 x 10^-11 N m^2/kg^2)
- m1 and m2 are the masses of the two bodies
- r is the distance between the centers of the two bodies

In this case, we want to find the gravitational force on a body due to the Earth at a height equal to half the radius of the Earth. Let's denote the mass of the body as m and the mass of the Earth as M.

2. Weight of the Body
The weight of a body is the force with which it is attracted towards the center of the Earth. On the surface of the Earth, the weight of a body can be calculated using the formula:

Weight = mass * acceleration due to gravity

Given that the body weighs 72 N on the surface of the Earth, we can equate this weight to the gravitational force between the body and the Earth:

Weight = F = G * (m * M) / r^2

3. Height Equal to Half the Radius of the Earth
At a height equal to half the radius of the Earth, the distance between the body and the center of the Earth can be calculated as follows:

Distance = radius of the Earth + height

Since we are considering half the radius, the distance can be written as:

Distance = (1/2) * radius of the Earth

4. Calculating the Gravitational Force
Using the given weight of the body and the relationship between weight and gravitational force, we can now solve for the gravitational force on the body at a height equal to half the radius of the Earth.

Weight = F = G * (m * M) / (1/2 * radius of the Earth)^2

Simplifying the equation, we can rearrange it to solve for the gravitational force:

F = Weight * (1/4) * (radius of the Earth)^2 / G

By substituting the given weight of 72 N and the appropriate values for the radius of the Earth and the gravitational constant, we can calculate the gravitational force on the body at the given height.

Conclusion
The gravitational force on a body at a height equal to half the radius of the Earth can be determined using the formula F = Weight * (1/4) * (radius of the Earth)^2 / G. By substituting the values, we can calculate the exact force.