This document ranks 4 hypothetical planets based on their mass and radius values and calculates the acceleration due to gravity (g) on each planet's surface. It determines that planet iii, with a mass 4 times Earth's and a radius 2 times Earth's, would have the highest g value of 9.8 m/s^2, equivalent to Earth's. Planet i, with double Earth's mass and radius, would have the second highest g value of 4.9 m/s^2. Planet ii, with quadruple Earth's mass and radius, would have the third highest g value of 2.45 m/s^2. Planet iv, with double Earth's mass but quadruple radius, would experience the lowest surface

1. Please solve this and explain it to me! Thanks
TEST YOUR UNDERSTANDING OF SECTION 13.2 Rank the following hypothetical planets
in order from highest to lowest value of g at the surface: (i) mass 2 times the mass of the earth,
radius = 2 times the radius of the earth; (ii) mass = 4 times the mass of the earth, radius 4 times
the radius of the earth; (iii) mass-4 times the mass of the earth, radius 2 times the radius of the
earth; (iv) mass = 2 times the mass of the earth, radius = 4 times the radius of the earth. I
Solution
Acceleration due to gravity on the earth's surface g = 9.8 m/s 2
Let the mass of the earth be M
Let the radius of the earth be R
Acceleration due to gravity at the earth's surface is given by
g = G M / R 2
............ ( 1 )
( i ) Here mass M1 = 2 M , radius R1 = 2 R
Then acceleration due to gravity g1 = G M1 / R1 2
g1 = G * 2 M / ( 2 R ) 2
= 2 G M / 4 R 2
= G M / 2 R 2
= g / 2
= 4.9 m/s 2
( ii ) Mass M2 = 4 M , radius R2 = 4 R

2. Acceleration due to gravity g2 = G * M2 / R2 2
g2 = G * 4 M / ( 4 R ) 2
= 4 G M / 16 R 2
= G M / 4 R 2
= g / 4
= 2.45 m/s 2
( iii ) Here Mass M3 = 4 M , radius R3 = 2 R
Acceleration due to gravity g3 = G * M3 / R3 2
g3 = G * 4 M / ( 2 R ) 2
= 4 G M /4 R 2
= G M / R 2
= g
= 9.8 m/s 2
( iv ) Here mass M4 = 2 M , radius R4 = 4R
Acceleration due to gravity g4 = G M4 / R4 2
g4 = G * 2 M / ( 4 R ) 2
= 2 G M / 16 R 2
= G M / 8 R 2
= g / 8
= 1.225 m/s 2
From above calculations , we can observe that case ( iii ) has highest value of g , while case ( iv )
have lowest value
Planets with g value in order of highest to lowest is
( iii ) , ( i ) , ( ii ) and ( iv )