$m_2$$ are the masses of the two objects, and $$r$$ is the distance between their centers.

The weight of the body on the surface of the Earth is given as $$100\,\text{N}$$, which is also the gravitational force acting on it:

$$F_\text{surface} = G \frac{m_\text{body} m_\text{earth}}{R_\text{earth}^2}$$

When the body is taken to a height equal to one-fourth the radius of the Earth, the distance between the centers of the body and the Earth becomes $$r_\text{new} = R_\text{earth} + \frac{1}{4} R_\text{earth} = \frac{5}{4} R_\text{earth}$$.

Now the gravitational force acting on the body at this new height is:

$$F_\text{new} = G \frac{m_\text{body} m_\text{earth}}{r_\text{new}^2}$$

$$F_\text{new} = G \frac{m_\text{body} m_\text{earth}}{\left(\frac{5}{4} R_\text{earth}\right)^2}$$

To find the ratio between the new gravitational force and the original force on the surface, we can write:

$$\frac{F_\text{new}}{F_\text{surface}} = \frac{G \frac{m_\text{body} m_\text{earth}}{\left(\frac{5}{4} R_\text{earth}\right)^2}}{G \frac{m_\text{body} m_\text{earth}}{R_\text{earth}^2}}$$

Canceling out the common terms, we get:

$$\frac{F_\text{new}}{F_\text{surface}} = \frac{R_\text{earth}^2}{\left(\frac{5}{4} R_\text{earth}\right)^2} = \frac{1}{\left(\frac{5}{4}\right)^2} = \frac{1}{\left(\frac{25}{16}\right)} = \frac{16}{25}$$

Now, since the weight of the body on the surface is $$100\,\text{N}$$, we can find the new gravitational force as:

$$F_\text{new} = \frac{16}{25} \times 100\,\text{N} = 64\,\text{N}$$

So, the gravitational force on the body when taken at a height equal to one-fourth the radius of the Earth is $$64\,\text{N}$$.

Gravitation

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