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Gravitation question

2023 · 1 Feb · Shift 2 · Q48
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  5. /2023 · 1 Feb · Shift 2 · Q48

Gravitation question

2023 · 1 Feb · Shift 2 · Q48

JEE MainPhysicsGravitationMCQ+4 / −1
The escape velocities of two planets A\mathrm{A}A and B\mathrm{B}B are in the ratio 1:21: 21:2. If the ratio of their radii respectively is 1:31: 31:3, then the ratio of acceleration due to gravity of planet A to the acceleration of gravity of planet B will be :
  1. A
    43\frac{4}{3}34​
  2. B
    23\frac{2}{3}32​
  3. C
    34\frac{3}{4}43​
  4. D
    32\frac{3}{2}23​
View written solutionFree

Correct answer: C

  1. For a planet, escape velocity is given by ve=2gRv_e = \sqrt{2gR}ve​=2gR​ where ggg is acceleration due to gravity and RRR is radius of the planet.

  2. Hence, ve2=2gR⇒g=ve22Rv_e^2 = 2gR \quad \Rightarrow \quad g = \frac{v_e^2}{2R}ve2​=2gR⇒g=2Rve2​​ So, g∝ve2Rg \propto \frac{v_e^2}{R}g∝Rve2​​

  3. Given: veA:veB=1:2v_{eA}:v_{eB} = 1:2veA​:veB​=1:2 RA:RB=1:3R_A:R_B = 1:3RA​:RB​=1:3

  4. Therefore,

= \frac{v_{eA}^2}{v_{eB}^2}\cdot \frac{R_B}{R_A}$$ Substitute the ratios: $$\frac{v_{eA}^2}{v_{eB}^2} = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$$ and $$\frac{R_B}{R_A} = \frac{3}{1} = 3$$ So, $$\frac{g_A}{g_B} = \frac{1}{4}\times 3 = \frac{3}{4}$$ 5. Hence, the required ratio is $$\boxed{\frac{3}{4}}$$ 6. Option check: - A: $\frac{4}{3}$ — incorrect - B: $\frac{2}{3}$ — incorrect - C: $\frac{3}{4}$ — correct - D: $\frac{3}{2}$ — incorrect
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