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 :  Two planets have radius in the ratio x:y and density in the ratio m:n  

To Find : The acceleration due to gravity on the surface of the two planets will be in the ratio  

Solution:

density = mass /Volume

=> Mass = density * volume

Volume = (4/3)πR³

Mass = d *  (4/3)πR³

g = GM/R²

=> g = G d *  (4/3)πR³ /R²

=> g =  (4Gπ/3) d.R

radius in the ratio x:y and  

=> Radius are xR  &  yR

density in the ratio m:n

=> densities are  md & nd

=> g₁ =  (4Gπ/3) md.xR

   g₂ =  (4Gπ/3) nd.yR

=> g₁ / g₂ =   mx/ny

=> g₁:g₂  = mx : ny

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17 Jul 2019 ... Know the answer

Answer 2

The ratio of acceleration due t gravity on the surface of the two planets will be xm : yn

Step-by-step explanation:

Given:

The ratio of the radius of two planets = x : y

The ratio of the densities of the planet = m : n

To find out:

The ratio of acceleration due to gravity

Solution:

We know that if the radius of a planet is R and mass M then the accleration due to gravity is given by

$g=\frac{GM}{R^2}$

if the density is d then

$M=V\times d$

$\implies M=\frac{4}{3}\pi R^3d$

Thus

$g=\frac{4}{3}\pi GRd$

If the acceleration due to gravity on the two planets is $g_1$ and $g_2$ , radii are $R_1$ and $R_2$ and densities are $d_1$ and $d_2$

Then

$\frac{R_1}{R_2}=\frac{x}{y}$

$\frac{d_1}{d_2}=\frac{m}{n}$

$g_1=\frac{4}{3}\pi GR_1d_1$

$g_2=\frac{4}{3}\pi GR_2d_2$

Therefore,

$\frac{g_1}{g_2}=\frac{R_1}{R_2}\times\frac{d_1}{d_2}$

$\implies \frac{g_1}{g_2}=\frac{xm}{yn}$

$\implies g_1:g_2=xm:yn$

Hope this answer is helpful.

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