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JEE Main 2019 (9 Apr, Shift 2) · Satellites - Orbital Velocity & Period

Satellites - Orbital Velocity & Period question from JEE Main 2019 (9 Apr, Shift 2)

Q61JEE Main 2019 · 9 APR · SHIFT 2Difficult

Satellites - Orbital Velocity & Period

A test particle is moving in a circular orbit in the gravitational field produced by a mass density ρ(r)=Kr2\rho(r) = \frac{K}{r^{2}}. Identify the correct relation between the radius R of the particle's orbit and its period T

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Answer: (d)

  1. Mass inside rr: M(r)=∫0rKr24πr2dr=4πKrM(r) = \int_0^r \frac{K}{r^2} 4\pi r^2 dr = 4\pi K r.
  2. v2R=G(4πKR)R2⇒v2=4πGK\frac{v^2}{R} = \frac{G(4\pi K R)}{R^2} \Rightarrow v^2 = 4\pi G K, so vv is constant.
  3. T=2πRv∝RT = \frac{2\pi R}{v} \propto R.
  4. So the answer is (d) T/RT/R is a constant.
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