Question

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 $6400 \,km$ :

• A. 1600 km
• B. 1400 km
• C. 2000 km
• D. 2600 km

Answer 1

The Correct Option is D

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 \( 2600 \, \text{km} \).