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Gravitation question

2023 · 12 Apr · Shift 1 · Q49
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  5. /2023 · 12 Apr · Shift 1 · Q49

Gravitation question

2023 · 12 Apr · Shift 1 · Q49

JEE MainPhysicsGravitationMCQ+4 / −1
Two satellites A\mathrm{A}A and B\mathrm{B}B move round the earth in the same orbit. The mass of A\mathrm{A}A is twice the mass of B\mathrm{B}B. The quantity which is same for the two satellites will be
  1. A
    Potential energy
  2. B
    Kinetic energy
  3. C
    Total energy
  4. D
    Speed
View written solutionFree

Correct answer: D

  1. Given

    • Two satellites AAA and BBB move around Earth in the same orbit.
    • Mass of satellite AAA is twice the mass of satellite BBB.
    • Let the common orbital radius be rrr.
  2. Speed of a satellite in a circular orbit The orbital speed is v=GMr,v = \sqrt{\frac{GM}{r}},v=rGM​​, where MMM is the mass of Earth.

    Since both satellites are in the same orbit, both have the same rrr. Also, GGG and MMM are constants. Therefore, vA=vB=GMr.v_A = v_B = \sqrt{\frac{GM}{r}}.vA​=vB​=rGM​​.

    So, speed is the same for both satellites.

  3. Check other options

    Let mass of satellite BBB be mmm. Then mass of AAA is 2m2m2m.

    (A) Potential energy

    Gravitational potential energy is U=−GMmr.U = -\frac{GMm}{r}.U=−rGMm​. For AAA: UA=−GM(2m)r=−2GMmr.U_A = -\frac{GM(2m)}{r} = -\frac{2GMm}{r}.UA​=−rGM(2m)​=−r2GMm​. For BBB: UB=−GMmr.U_B = -\frac{GMm}{r}.UB​=−rGMm​. Hence, UA≠UBU_A \ne U_BUA​=UB​.

    (B) Kinetic energy

    K=12mv2=GMm2r.K = \frac{1}{2}mv^2 = \frac{GMm}{2r}.K=21​mv2=2rGMm​. For AAA: KA=GM(2m)2r=GMmr.K_A = \frac{GM(2m)}{2r} = \frac{GMm}{r}.KA​=2rGM(2m)​=rGMm​. For BBB: KB=GMm2r.K_B = \frac{GMm}{2r}.KB​=2rGMm​. Hence, KA≠KBK_A \ne K_BKA​=KB​.

    (C) Total energy

    For a satellite in circular orbit, E=K+U=−GMm2r.E = K + U = -\frac{GMm}{2r}.E=K+U=−2rGMm​. For AAA: EA=−GM(2m)2r=−GMmr.E_A = -\frac{GM(2m)}{2r} = -\frac{GMm}{r}.EA​=−2rGM(2m)​=−rGMm​. For BBB: EB=−GMm2r.E_B = -\frac{GMm}{2r}.EB​=−2rGMm​. Hence, EA≠EBE_A \ne E_BEA​=EB​.

  4. Conclusion The only quantity same for both satellites is: Speed\boxed{\text{Speed}}Speed​

  5. Comparison with stored correct answer Stored correct answer: DDD

    Our derived answer: DDD

    So, the derived answer agrees with the stored correct answer.

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