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Gravitation question

2021 · 25 Feb · Shift 1 · Q63
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  5. /2021 · 25 Feb · Shift 1 · Q63

Gravitation question

2021 · 25 Feb · Shift 1 · Q63

JEE MainPhysicsGravitationMCQ+4 / −1
Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R. Assertion A : The escape velocities of planet A and B are same. But A and B are of unequal mass. Reason R : The product of their mass and radius must be same. M1R1 = M2R2 In the light of the above statements, choose the most appropriate answer from the options given below :
  1. A
    Both A and R are correct and R is the correct explanation of A
  2. B
    Both A and R are correct but R is NOT the correct explanation of A
  3. C
    A is correct but R is not correct
  4. D
    A is not correct but R is correct
View written solutionFree

Correct answer: C

  1. Escape velocity formula

    For a planet of mass MMM and radius RRR, the escape velocity is ve=2GMRv_e = \sqrt{\frac{2GM}{R}}ve​=R2GM​​ where GGG is the gravitational constant.

  2. Given assertion

    The assertion says that planets AAA and BBB have the same escape velocity, but unequal masses.

    If their escape velocities are same, then 2GM1R1=2GM2R2\sqrt{\frac{2GM_1}{R_1}} = \sqrt{\frac{2GM_2}{R_2}}R1​2GM1​​​=R2​2GM2​​​

    Squaring both sides, M1R1=M2R2\frac{M_1}{R_1} = \frac{M_2}{R_2}R1​M1​​=R2​M2​​

    So equal escape velocities require M1R2=M2R1M_1R_2 = M_2R_1M1​R2​=M2​R1​

    This does not require M1=M2M_1 = M_2M1​=M2​. Hence planets can have the same escape velocity even if their masses are unequal, provided their radii are also different appropriately.

    Therefore, Assertion A is correct.

  3. Check the Reason

    Reason says: M1R1=M2R2M_1R_1 = M_2R_2M1​R1​=M2​R2​

    But from the escape velocity condition, we obtained M1R1=M2R2\frac{M_1}{R_1} = \frac{M_2}{R_2}R1​M1​​=R2​M2​​ or equivalently, M1R2=M2R1M_1R_2 = M_2R_1M1​R2​=M2​R1​

    This is not the same as M1R1=M2R2M_1R_1 = M_2R_2M1​R1​=M2​R2​

    So the reason is incorrect.

  4. Conclusion

    • Assertion A: correct
    • Reason R: incorrect

    Therefore, the correct option is: C\boxed{\text{C}}C​

  5. Comparison with stored answer

    Stored correct answer is C, which matches the derived answer.

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