Question

the acceleration due to gravity of planet in g upon 8 . if the radius is half of earths radius. then the ratio mass of planet to mass of earth is

1 : 32
1 : 8
1 : 16
1 : 64​

Answer 1

Answer :-

The ratio of mass of the planet to mass of Earth is 1 : 32 . [Option.1]

Explanation :-

For Earth :-

• Mass = Mₑ

• Radius = r

• Acceleration due to gravity = g

So the equation for acceleration due to gravity becomes :-

⇒ g = GMₑ/r²

⇒ gr² = GMₑ ------(1)

For the other planet :-

• Mass = Mₚ

• Radius = r/2

• Acceleration due to gravity = g/8

The equation for acceleration due to gravity :-

⇒ g/8 = GMₚ/(r/2)²

⇒ g/8 = 4GMₚ/r²

⇒ g(r)² = 8(4GMₚ)

⇒ gr² = 32GMₚ -----(2)

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On dividing eq.2 by eq.1, we get :-

⇒ gr²/gr² = 32GMₚ/GMₑ

⇒ 1 = 32Mₚ/Mₑ

⇒ Mₑ = 32Mₚ

⇒ Mₚ = Mₑ/32

⇒ Mₚ : Mₑ = 1 : 32

Answer 2

Given :-

the acceleration due to gravity of planet in g upon 8 . if the radius is half of earths radius.

To Find :-

Mass of planet to earth

Solution :-

Let us assume that

Mass of Earth = M

Mass of Planet = M'

Since,  acceleration due to gravity of planet id g/8

So,

For other planet

Radius = Radius of Earth/2

g/8 =  GM'/[r/2]²

g/8 = GM'/r²/4

g/8 = GM'/r² × 4

g/8 = 4GM'/r²

g × r² = 8 × 4GM'

gr² = 32GM'

For Earth

g =  GM/r²

g × r² = GM

gr² = GM

gr²/gr² = 32GM'/GM

g/g = 32GM'/GM

1 = 32GM'/GM

1 = 32M'/M

1(M) = 32M'

M = 32M'

M/32M'

Ratio = 1:32