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MCQEasyJEE 2023Acceleration due to Gravity

JEE Physics 2023 Question with Solution

Two planets A and B of radii RR and 1.5R1.5R have densities ρ\rho and ρ/2\rho/2 respectively. The ratio of acceleration due to gravity at the surface of B to A is:

  • A

    2:32 : 3

  • B

    2:12 : 1

  • C

    4:34 : 3

  • D

    3:43 : 4

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: Planet A has radius RR and density ρ\rho. Planet B has radius 1.5R1.5R and density ρ/2\rho/2.

Find: The ratio of acceleration due to gravity at the surface of B to A.

For a planet,

g=GMR2g = \frac{GM}{R^2}

and using

M=43πR3ρM = \frac{4}{3}\pi R^3 \rho

we get

g=G⋅43πR3ρR2=43πGRρg = \frac{G \cdot \frac{4}{3}\pi R^3 \rho}{R^2} = \frac{4}{3}\pi G R \rho

Using proportionality

Since

g∝Rρg \propto R\rho

we have

gBgA=RBρBRAρA\frac{g_B}{g_A} = \frac{R_B\rho_B}{R_A\rho_A}

Substituting

RB=1.5R,ρB=ρ2,RA=R,ρA=ρR_B = 1.5R, \quad \rho_B = \frac{\rho}{2}, \quad R_A = R, \quad \rho_A = \rho

so

gBgA=(1.5R)(ρ2)Rρ=1.52=34\frac{g_B}{g_A} = \frac{(1.5R)\left(\frac{\rho}{2}\right)}{R\rho} = \frac{1.5}{2} = \frac{3}{4}

Therefore, the ratio of acceleration due to gravity at the surface of B to A is 3:43:4. Hence, the correct option is D.

The solution also contains an inconsistent first approach where the heading says option C, but the worked value concludes 3:43:4. The final worked result matches option D, so D is correct.

Common mistakes

  • Using g∝R2ρg \propto R^2\rho instead of g∝Rρg \propto R\rho. This is wrong because after substituting M=43πR3ρM = \frac{4}{3}\pi R^3\rho into g=GMR2g = \frac{GM}{R^2}, one power of RR remains. Always simplify the formula fully before taking ratios.

  • Taking the ratio as gA/gBg_A/g_B instead of gB/gAg_B/g_A. This reverses the final answer. Read carefully which planet is asked in the numerator and which in the denominator.

  • Trusting the option label written in the solution heading without checking the working. Here the heading says option C, but the actual calculation gives 3:43:4. Follow the derivation, not the mislabeled heading.

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