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Q. 68

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
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Short Answer

A satellite in a circular orbit of radius r has period T. A satellite in a nearby orbit with radius r + Δr, where Δr << r , has the very slightly different period T+ ΔT.

a) Show that

b) Two earth satellites are in parallel orbits with radii 6700 km and 6701 km. One day they pass each other, 1 km apart, along a line radially outward from the earth. How long will it be until they are again 1 km apart?

a) The given expression is proved.

b) They will meet again after 281.05 days

See the step by step solution

Step by Step Solution

Part(a) Step1: Given information

Two satellites are crossing each other. r and T are the radius of orbit and time period of first satellite whereas r + Δr, and T + ΔT, are the radius of orbit and time period of the second satellite respectively.

Part(a)Step2: Explanation

From Kepler's law

Now lets assume , the first satellite obeys the .
And , for the second satellite as it varies with Δr and ΔT so it will be

as

so,

Now subtracting the equation for the first satellite:

On dividing with the equation for T, we get

And proved.

Part(b) Step1: Given information

Two satellites are in parallel orbit.
Radius of orbit of first satellite,r=6700 kmRadius of orbit of second satellite, r + Δr=6701 kmMass of Earth =5.98 x 1024 kg

Part(b)Step2: Explanation

In part(a) of the problem we have established

Substitute values

So after periods, they will meet again.

Find the value of One period

So they will meet again after 4467 periods = 4467 x 1.51 hours =6745.2 hours

.6745.2 hours =281.05 days .

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