Suppose there are two planets, 1 and 2, having the same density but th

Suppose there are two planets, 1 and 2, having the same density but their radii are R₁ and R₂ respectively, where R₁ > R₂. The accelerations due to gravity on the surface of these planets are related as

Join Our Telegram Channel

[amp_mcq option1=”g₁ > g₂” option2=”g₁ < g₂" option3="g₁ = g₂" option4="Can't say anything" correct="option1"]

This question was previously asked in

UPSC NDA-1 – 2019

Download PDF Attempt Online

If two planets have the same density but different radii R₁ and R₂ with R₁ > R₂, the acceleration due to gravity on the surface of the planet with the larger radius (Planet 1) will be greater than that on the surface of the planet with the smaller radius (Planet 2), i.e., g₁ > g₂.

– The acceleration due to gravity on the surface of a sphere is given by g = GM/R², where G is the gravitational constant, M is the mass, and R is the radius.
– The mass M can be expressed as M = ρ * V, where ρ is the density and V is the volume. For a sphere, V = (4/3)πR³.
– Substituting this into the gravity formula: g = G * (ρ * (4/3)πR³) / R² = (4/3)πGρR.
– Since the density ρ and constants G, 4/3, and π are the same for both planets, the acceleration due to gravity is directly proportional to the radius (g ∝ R).
– Given R₁ > R₂, it follows that g₁ > g₂.

This relationship shows that for objects of the same density, larger objects exert stronger gravitational pull at their surface compared to smaller objects. This is because the increase in mass (proportional to R³) outpaces the increase in distance from the center (proportional to R²).

More similar MCQ questions for Exam preparation:-