Physics

Physics

Problem

Two spherical planets A and B are made of the same uniform material of density ρ\rho. Planet A has radius 2R2R while planet B has radius RR. a. (i) Express the mass of each planet in terms of ρ\rho and its radius. b. (ii) Using your results from part (i), obtain the ratio of the minimum speeds required to escape from the surfaces of planets A and B. c. (iii) Compute the numerical value of this ratio.
Intuitive explanation

The key is recognizing that gravitational force depends on mass, which scales with volume. Since both planets have the same density, the larger planet A has 23=82^3 = 8 times more mass than B. This means the surface gravity on each planet—determined by how much mass pulls on you divided by how far you are from that mass—will differ not just because of the different masses, but also because you're standing at different distances from their centers. Work out how surface gravity g=GMr2g = \frac{GM}{r^2} scales with radius when density is held constant.

Intuitive explanation

The key is recognizing that gravitational force depends on mass, which scales with volume. Since both planets have the same density, the larger planet A has 23=82^3 = 8 times more mass than B. This means the surface gravity on each planet—determined by how much mass pulls on you divided by how far you are from that mass—will differ not just because of the different masses, but also because you're standing at different distances from their centers. Work out how surface gravity g=GMr2g = \frac{GM}{r^2} scales with radius when density is held constant.

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