Gravitation Part 1

Let the mass of planet be mp and that of earth be m .

Given,
mp=2m

Density of planet (ρp)= Density of
earth (ρ)
As we know that,
Weight of object on earth is W
w=mg‌‌‌⋅⋅⋅⋅⋅⋅⋅(i)
where, g is acceleration due to gravity.

and ‌‌g=‌

Gm

R2

where, ‌‌G= universal gravitational
constant,
R‌‌=‌ radius of earth ‌
‌ and ‌‌‌me‌‌=‌ mass of earth. ‌
⇒g‌‌=‌

Gρ 4 3 πR3

R2

⇒g‌‌=Gρ‌

4

3

πR
Putting this value in Eq. (i), we get
w=mGρ‌

4

3

πR‌‌‌⋅⋅⋅⋅⋅⋅⋅(ii)

As densities of planets earth is same.
i.e., ‌‌ρp=ρ
⇒‌

mp

4 3 πRp3

‌‌=‌

m

4 3 πR3

⇒‌‌Rp‌‌=(‌

mp

m

R3)1∕3
‌‌=(‌

2m

m

R3)1∕3
⇒‌‌Rp‌‌=(2)1∕3R
Weight on planet,
wp=‌

4

3

(2)1∕3πmGρR‌‌‌⋅⋅⋅⋅⋅⋅⋅(iii)
Dividing Eq. (iii) by Eq. (ii), we get
‌

wp

w

‌‌=‌

‌ 4 3 (2)1∕3πmGρR

mρG‌ 4 3 πR

⇒‌‌wp‌‌=21∕3w