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Gravitation question

2023 · 10 Apr · Shift 1 · Q58
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  5. /2023 · 10 Apr · Shift 1 · Q58

Gravitation question

2023 · 10 Apr · Shift 1 · Q58

JEE MainPhysicsGravitationMCQ+4 / −1
Two satellites of masses m and 3m revolve around the earth in circular orbits of radii r & 3r respectively. The ratio of orbital speeds of the satellites respectively is
  1. A
    3 : 1
  2. B
    3\sqrt33​ : 1
  3. C
    1 : 1
  4. D
    9 : 1
View written solutionFree

Correct answer: B

  1. Use the formula for orbital speed

For a satellite revolving in a circular orbit around the earth, the orbital speed is

v=GMRv = \sqrt{\frac{GM}{R}}v=RGM​​

where:

  • GGG is the gravitational constant,
  • MMM is the mass of the earth,
  • RRR is the orbital radius.

Notice that the orbital speed is independent of the mass of the satellite.

  1. Speed of the first satellite

For the satellite of mass mmm in orbit of radius rrr,

v1=GMrv_1 = \sqrt{\frac{GM}{r}}v1​=rGM​​

  1. Speed of the second satellite

For the satellite of mass 3m3m3m in orbit of radius 3r3r3r,

v2=GM3rv_2 = \sqrt{\frac{GM}{3r}}v2​=3rGM​​

  1. Find the ratio

v1v2=GM/rGM/(3r)=3\frac{v_1}{v_2} = \frac{\sqrt{GM/r}}{\sqrt{GM/(3r)}} = \sqrt{3}v2​v1​​=GM/(3r)​GM/r​​=3​

So,

v1:v2=3:1v_1 : v_2 = \sqrt{3} : 1v1​:v2​=3​:1

  1. Check options
  • A: 3:13:13:1 ❌
  • B: 3:1\sqrt{3}:13​:1 ✅
  • C: 1:11:11:1 ❌
  • D: 9:19:19:1 ❌

Therefore, the correct option is B.

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