The ratio of masses of two planets is 2:3 and the ratio of their radii is 4:7 . Find the ratio of their acceleration due to gravity .

Explanation

Given
The radio of masses of two bodies is `m_1:m_2`
Mass of the smaller body `, m_1=2 kg `
Mass of the bigger body `m_2=3 kg `
Radius of the smaller body , `R_1=4 km`
Radius of the bigger body , `R_2=7 km `
i.e `r_1:r_2=4:7`
To find : `g_1 , g_2=?`
Solution
We know that
`g=(Gm)/(R^(2))`
`g_1=(GM_1)/(R_1^(2)), g_2=(GM_2)/(R_2^(2))`
`(g_1)/(g_2) =((GM_1)/(R_1^(2)))/((GM_2)/(R_2^(2)))=(M_1)/(R_1^(2))xx(R_2^(2))/(M_2)`
`(g_1)/(g_2)=((M_1)/(M_2))((R_2)/(R_1))^(2)=((2)/(3))((7)/(4))^(2)`
`(g_1)/(g_2)=(2)/(3)xx(7 xx 7)/(cancel(4_2) xx 4)=(49)/(24)`
The ratio is , `g_1:g_2=49:24`