At what height from the surface of earth the gravitation potential and the value of g are $- 5.4 \times 10^7 \, J \, kg^{-2}$ and $6.0 \, ms^{-2}$ respectively ? Take the radius of earth as $6400 \,km$ :
- 1600 km
- 1400 km
- 2000 km
- 2600 km
The Correct Option is D
Solution and Explanation
Gravitational Potential and Height Calculation
The gravitational potential \( V \) at a distance \( R + h \) is given by the formula:
\(V = \frac{-GM}{R + h} = -5.4 \times 10^{7}\) ....(1)
The gravitational field strength \( g \) at the same distance is given by the formula:
\(g = \frac{GM}{(R + h)^2} = 6\) ......(2)
Dividing Equations (1) and (2):
By dividing equation (1) by equation (2), we get:
\(\frac{5.4 \times 10^{7}}{(R + h)} = 6\)
Solving for \( R + h \):
\(R + h = \frac{5.4 \times 10^{7}}{6} = 9000 \, \text{km}\)
Therefore, the height \( h \) is:
\(h = 9000 \, \text{km} - R\)
Given that the radius of the Earth \( R \) is approximately 6400 km, we have:
\(h = 9000 \, \text{km} - 6400 \, \text{km} = 2600 \, \text{km}\)
Conclusion:
The height \( h \) is approximately \( 2600 \, \text{km} \).
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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].