Animated Solution for Physics - Gravitation: The ratio of the weights of a body on the earth's surface, so that on the surface of a planet is 9 : 4. The mass of the planet is $\frac{1}{9}$th of that of the earth. If $R$ is the radius of the earth, what is the radius of the planet? (Take, the planets to have the same mass density)

You first place it on the surface of the Earth and measure its weight, $W_e$. Then, you take this exact same body to a mysterious new planet and measure its weight again, $W_p$. The problem tells us that the ratio of these weights is $9:4$.

What exactly is weight? Weight is simply the gravitational force exerted by a planet on an object at its surface. Mathematically, it is the product of the object's mass and the acceleration due to gravity, $W=mg$.

Since the mass $m$ of the body remains constant regardless of where it is in the universe, the ratio of the weights is directly equal to the ratio of the acceleration due to gravity on the two planets:

It is given by Newton's law of universal gravitation:

where $G$ is the universal gravitational constant, $M$ is the mass of the planet, and $R$ is its radius. Let's write this expression for both the Earth and our mystery planet.

For Earth: $g_e=\frac{GM}{R^2}$

For the Planet: $g_p=\frac{G{M}_p}{R_p^2}$

Let's substitute these expressions back into our ratio:

Notice how the universal gravitational constant $G$ beautifully cancels out. Rearranging the terms, we get a relation involving the ratios of their masses and the square of the ratio of their radii:

We are given a crucial piece of information: the mass of the planet is $\frac{1}{9}$th the mass of the Earth.

Substituting this mass ratio into our equation, we get:

The $9$ on both sides cancels out perfectly, leaving us with:

Taking the square root of both sides (and keeping only the positive root since radius is a physical distance), we find:

Therefore, the radius of the planet is exactly half the radius of the Earth.

A Note on the Problem Statement: You might have noticed the phrase "Take, the planets to have the same mass density" at the end of the question. If the densities were truly the same, the mass would be proportional to the cube of the radius ($M\propto R^3$), which would mean $R_p=R/9^{1/3}$. This contradicts the given options and the intended solution path. In competitive exams, such contradictory statements occasionally appear, and it is crucial to follow the primary numerical data (the mass ratio) to arrive at the correct option.