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A satellite can be in a geostationary orbit around the earth at a distance r from the center. If the angular velocity of the earth about its axis doubles, a satellite can now be in a geostationary orbit around the earth if its distance from the center is,


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Solution

Angular velocity

  1. The angular velocity is the time rate of change of angular displacement of a moving body.
  2. The angular velocity of the satellite is defined by the form, ω=GMr3, where, r is the distance of the satellite from the earth, G is the universal gravitational constant, and M is the mass of the satellite.
  3. The angular velocity of a geostationary satellite is the same as the earth's angular velocity.

Step 1: Given data

  1. The distance of the satellite from the earth's center is r.

Step 2: Finding the distance

Let the distance of the satellite after increasing angular velocity is R.

Initially, the angular velocity of the satellite is

ωi=GMr3.............(1)

Finally, the angular velocity is doubled. Now angular velocity is,

2ωi=GMR3.............(2)

Comparing, equations 1 and 2 we get,

GMr3= 12GMR3

or1r3=141R3or4R3=r3orR3=r34orR=r413

Therefore, the distance of the satellite after increasing angular velocity is r413.


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