Question:
A remote-sensing satellite of earth revolves in a circular orbit at a height of $0.25 \times 10^6\, m$ above the surface of earth. If earth's radius is $6.38 \times 10^6$ m and $g = 9.8 \,ms^{-2}$, then the orbital speed of the satellite is
A remote-sensing satellite of earth revolves in a circular orbit at a height of $0.25 \times 10^6\, m$ above the surface of earth. If earth's radius is $6.38 \times 10^6$ m and $g = 9.8 \,ms^{-2}$, then the orbital speed of the satellite is
Updated On: May 21, 2024
- $9.13 \,km\,s^{-1}$
- $6.67 \, km\,s^{-1}$
- $7.76 \, km\,s^{-1}$
- $8.56 \, km\,s^{-1}$
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The Correct Option is C
Solution and Explanation
The orbital speed of the satellite is
$V_o = R \sqrt \frac {g}{(R+h)}$
where R is the earth's radius, g is the acceleration due to gravity on earth's surface and h is the height above the surface of earth.
Here, $R = 6.38 \times 10^6m$, $\, g = 9.8 \,ms^{-2}$ and
$h = 0.25 \times 10^6\, m$
$\therefore V_o (6.38 \times 10^6\,m) \sqrt \frac {(9.8 \, ms^{-2})}{(6.38\, \times \,10^6 \, m + 0.25\, \times\, 10^6 \,m)}$
$ 7.76 \times 10^3 \,ms^{-1} = 7.76 \, kms^{-1}$ $ (\therefore 1\,km = 10^3\, m)$
$V_o = R \sqrt \frac {g}{(R+h)}$
where R is the earth's radius, g is the acceleration due to gravity on earth's surface and h is the height above the surface of earth.
Here, $R = 6.38 \times 10^6m$, $\, g = 9.8 \,ms^{-2}$ and
$h = 0.25 \times 10^6\, m$
$\therefore V_o (6.38 \times 10^6\,m) \sqrt \frac {(9.8 \, ms^{-2})}{(6.38\, \times \,10^6 \, m + 0.25\, \times\, 10^6 \,m)}$
$ 7.76 \times 10^3 \,ms^{-1} = 7.76 \, kms^{-1}$ $ (\therefore 1\,km = 10^3\, m)$
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Concepts Used:
Gravitation
In mechanics, the universal force of attraction acting between all matter is known as Gravity, also called gravitation, . It is the weakest known force in nature.
Newton’s Law of Gravitation
According to Newton’s law of gravitation, “Every particle in the universe attracts every other particle with a force whose magnitude is,
- F ∝ (M1M2) . . . . (1)
- (F ∝ 1/r2) . . . . (2)
On combining equations (1) and (2) we get,
F ∝ M1M2/r2
F = G × [M1M2]/r2 . . . . (7)
Or, f(r) = GM1M2/r2
The dimension formula of G is [M-1L3T-2].