Question

Two planets have radius in the ratio x : y and density in the ratio m : n. The acceleration due to gravity on the surface of the two planets will be in the ratio of

Answer 1

Given :-

◉$\quad$ Two planets have radius in the ratio x : y and density in ratio m : n.

To Find :-

◉$\quad$ The ratio of acceleration due to gravity on the surface of the two planets will be?

Solution :-

We know,

$\qquad \displaystyle \boxed{ \rm g = \frac{Gm}{ {r}^{2} } }$

From the equation, we know

$\rightarrow \qquad \displaystyle \rm g \propto \frac{1}{ {r}^{2} } \\ \\ \displaystyle \rm and, \quad g \propto M$

It is given that the ratio of density of these two planets is m : n

So, Let us find the ratio of masses,

$\Rightarrow \displaystyle \quad \rm \frac{Density_{a}}{Density_{b}} = \frac{m}{n} \\ \\ \Rightarrow \displaystyle \rm \quad \frac{ \frac{m_{a}}{v_{a}}}{ \frac{m _{b}}{v_{b}}} = \frac{m}{n} \\ \\ \Rightarrow \displaystyle \rm \quad \frac{m_{a} \cdot v_{b}}{m_{b} \cdot v_{a}} = \frac{m}{n} \\ \\ \Rightarrow \displaystyle \rm \quad \frac{m_{a}}{m_{b}} = \frac{m \cdot v_{a}}{n \cdot v_{b}} \quad ...(1)$

Now, that we have found the ratio of thier masses, Let us find the ratio of acceleration due to gravity:

$\displaystyle \Rightarrow \quad \rm \frac{g_{a}}{g_{b}} = \frac{ \frac{\not{G}m_{a}}{ {(r _{a})}^{2} } }{ \frac{\not{G}m_{b}}{ {(r_{b})}^{2} } } \\ \\ \Rightarrow \displaystyle \rm \quad \frac{g_{a}}{g_{b}} = \frac{m_{a} \cdot {(r_{b})}^{2} }{m _{b} \cdot {(r_{a})}^{2} } \\ \\ \Rightarrow \displaystyle \quad \rm \frac{g _{a}}{g_{b}} = \frac{m\cdot v_{a}}{n \cdot v_{b}} \times \frac{ {y}^{2} }{ {x}^{2} } \\ \\ \Rightarrow \displaystyle \quad \rm \frac{g _{a}}{g_{b}} = \frac{ {y}^{2}mv_{a}}{ {x}^{2}nv_{b}} \\ \\ \Rightarrow \displaystyle \quad \rm \frac{g_{a}}{g_{b}} = \frac{y^{2} \cdot m \cdot \not{4}\not{\pi}(r_{a})^{3}}{x^{2} \cdot n \cdot \not{4}\not{\pi}(r_{b})^{3}}$$\Rightarrow \displaystyle \quad \rm \frac{g_{a}}{g_{b}} = \frac{x^{\not{3}^{1}}y^{\not{2}}m}{y^{\not{3}_{1}}x^{\not{2}}n} \\ \\ \Rightarrow \displaystyle \quad \rm \frac{g_{a}}{g_{b}} = \frac{xm}{yn}$

Some Information :-

◉$\quad$Volume of Sphere = 4πr³

Where, r = radius of sphere

◉$\quad$ Mass = Density × Volume