Question

Imagine a new planet having the same density as that of earth but it is 3 times bigger than the earth in size. If the acceleration due to gravity on the surface of earth is g and that on the surface of the new planet is g' then what would be the relation between them?

Answer 1

Step 1: Acceleration due to gravity

Suppose an object falls freely towards the surface of the earth, then its velocity gets changed and this change produces acceleration for a particular object which is referred to as acceleration that occurs due to gravity.

It is denoted by the symbol $g$. The value of $g$ is $9.8m/s^2$.

Step 2: Relative Radius

If the mass and radius of the new planet be $M\text{'}$ and $R\text{'}$, then for same density as earth

$R\text{'}=3R$

From the above statement, we know that,

Mass $\left(m\right)$ is directly proportional to velocity $\left(v\right)$

Where $v$ is directly proportional to $R^3$

Step 3: Establish the Relative

Now for Earth,

$g=\frac{GM}{R^2}$ and $g\text{'}=\frac{GM\text{'}}{R{\text{'}}^2}$

Now we get $g\text{'}=\frac{G\times 3M}{{\left(3R\right)}^2}$

Taking $3$ common from numerator and denominator we get,

$g\text{'}=\frac{GM}{3R^2}$

$3g\text{'}=g$

Thus, the relation between them is $3g\text{'}=g$.