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 )