Calculate the energy required to move an earth satellite of mass `10^(

1.	Calculate the energy required to move an earth satellite of mass `10^(3)` kg from a circular orbit of radius 2R to that of radius 3R. Given mass of the earth,`M=5.98xx10^(24)` kg and radius of the earth, `R=6.37xx10^(6)`m.
Answer» Given M = 5.98 x 10²⁴ kg m = 10³ kg R = 6.37 x 10⁶ m Centripetal acceleration of the satellite is provided by the gravitational force exerted by Earth. \(\frac{mv^2}{r}=\frac{GMm}{r^2}\) v² = GM/r Total energy of the system = K.E. + Gravitational Potential energy = 1/2 mv² - GMm/r = 1/2 m GM/r - GMm/r = - 1/2 GMm/r Therefore, energy required to move on Earth satellite radius 2R to orbital radius 3R. E = GMm/2 (1/2R - 1/3R) E = GMm/12R \(E=\frac{6.67\times10^{-11}\times5.98\times10^{24}\times10^3}{12\times6.37\times10^6}\) \(E=\frac{39.88\times10^{10}}{76.44}\) ⇒ 0.5217 x 10¹⁰ J Correct Answer - `5.02xx10^(9)`J

Calculate the energy required to move an earth satellite of mass `10^(3)` kg from a circular orbit of radius 2R to that of radius 3R. Given mass of the earth,`M=5.98xx10^(24)` kg and radius of the earth, `R=6.37xx10^(6)`m.

Answer»

Given M = 5.98 x 10²⁴ kg

m = 10³ kg

R = 6.37 x 10⁶ m

Centripetal acceleration of the satellite is provided by the gravitational force exerted by Earth.

\(\frac{mv^2}{r}=\frac{GMm}{r^2}\)

v² = GM/r

Total energy of the system = K.E. + Gravitational Potential energy

= 1/2 mv² - GMm/r

= 1/2 m GM/r - GMm/r

= - 1/2 GMm/r

Therefore, energy required to move on Earth satellite radius 2R to orbital radius 3R.

E = GMm/2 (1/2R - 1/3R)

E = GMm/12R

\(E=\frac{6.67\times10^{-11}\times5.98\times10^{24}\times10^3}{12\times6.37\times10^6}\)

\(E=\frac{39.88\times10^{10}}{76.44}\)

⇒ 0.5217 x 10¹⁰ J

Correct Answer - `5.02xx10^(9)`J

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