Question

At what height from the surface of the earth the value of the gravitational potential and the value of the g are: $-5.4 \times 10^7$ and $6 ms^{-2}$ respectively. Take the radius of the earth is 6400 km.
A) 2600 km
B) 1600 km
C) 1400 km
D) 2000 km

Answer 1

Hint: Use the formula for the gravitational potential and find the equation from the given value of the potential and then use the formula for the g value and then obtain another equation using the given value of the g.
Now divide the two equations and find the value of the h height.

Complete step by step solution:
Let's assume the required value of the height as h
The value of the gravitational potential can be obtained using the formula as
$V = \dfrac{-GM}{R+h}$
Where the G is the gravitational constant
M is the mass of the earth
m is the mass of the body
R is the radius of the earth and now
The value of the given potential $V$ at height is $-5.4 \times 10^7$
Now using the value of the potential and the definition of the potential we can write the equation as
$V = \dfrac{-GM}{R+h}$ = $-5.4 \times 10^7$ $ms^{-2}$
Now the value of the g given is 6 $ms^{-2}$
We know the g value at height h is given as
$g = \dfrac{GM}{(R+h)^2}$
Where the G is the gravitational constant
M is the mass of the earth
m is the mass of the body
R is the radius of the earth and now
We can write the equation using the given value of the potential and the value of the g defined above as
$g = \dfrac{GM}{(R+h)^2}$ $=$ $6 ms^{-2}$
Hence now we have obtained two equations here and we need to find the value of height h
We can do one thing. We can divide both the equations and then find the value of the unknown quantity h. The LHS and RHS of the two equations are divided and we get the value of the LHS and RHS after the division as below
$\dfrac{5.4 \times 10^7}{R+h} = 6$
Now we can find the value of h as
$R+h = 9000 km$
h = 9000-6400 = 2600 km
Thus we have used the formula for the gravitational potential and the formula for the value of the g and we got the required value of the height as: 2600 km

Note: One of the possible mistakes that we can make in this kind of problem is that we have to be careful about the units while substituting the given values. Take care that the value of R is given in km and we need to convert it in meters to proceed further.