Question

Two small satellites move in circular orbits around the earth,at distance r and r+delta r from the centre of earth their time periods of rotation T and T+deltaT [delta r <<

Answer 1

Your question ,

• Two small satellites move in circular orbits around the earth,at distance r and r+delta r from the centre of earth their time periods of rotation T and T+deltaT [delta r <<

Correct question may be ,

• Two small satellites move in circular orbits around the earth,at distance r and r+∆ r from the centre of earth their time periods of rotation T and T+∆T [delta r <<< r and ∆T <<< T ]

Here if you want the the derivation for ∆T

Then ,

Your Answer is ,

Given in the question ,

• Two satellites circulating in their orbits around the earth.
• The given distances from the centre of the earth to the orbits are r and r+∆r.
• Time periods of rotation for given satellites are T and T + ∆T.

Here , we can find the ∆T 's value

As we can say that , T ² directly proportional to the r³

So,

• T² = cr³ .............( 1 )
• we can express c as a constant.

Also from the question , we can say that ,

For the second satellite

• ( T + ∆T )² = c ( r + ∆ r )³

if r >>> ∆r and T >>> ∆T

Then ,

• T² + 2T∆T = c ( r³ + 3r²∆r )

in this equation we neglected ∆T² and ∆r³as r >>>∆r and T >>> ∆T ( Already discussed ).

• T² + 2T∆T = cr³ + c3r²∆r

By , putting the value of equation ( 1 ) in this equation , we get ,

• cr³ + 2T∆T = cr³ + c3r²∆r
• 2T∆T = c3r²∆r. ( new )

This new equation , when devided by equation ( 1 ) , we get ,

• 2T∆T / T² = c3r²∆r / cr³
• 2∆T / T = 3∆r / r

So ,

• ∆T = 3T∆r / 2r

→ We also can find the derivation for r and T now from same method