The gravitational potential of the earth acting on a mass m

The gravitational potential of the earth acting on a mass m is V(r) = -GM_0m/r, where M_0 is the mass of the earth. If the radius of the earth is R_0, find the gravitational force mg acting on a mass at the earth\'s surface in terms of R0. The escape velocity vQ is defined as that velocity for which the total energy (kinetic plus potential) is zero. Find v0 as a function of R0 and g. Now, assuming that R_0 ~ 6 Times 10^3 km, and g ~ 10 ms^-2, find an approximate answer for v_0 for the earth (you will not need a calculator for this!!).

Solution

The force of gravity (F_g) with which an object of mass m was attracted to the earth

F_g = m*g ... (1)

Now, a second equation has been introduced for calculating the force of gravity with which an object is attracted to the earth.

F_g= G*M₀*m/ (R₀²) ... (2)

Where, G = 6.67 * 10^-11 units.

Equating equations (1) and (2), m cancels out and we get:

g = G*M₀/ (R₀²)

The above equation demonstrates that the acceleration of gravity is dependent upon the mass of the earth (approx. 5.98x10^24 kg) and the distance (R₀) that an object is from the center of the earth. If the value 6.38x10^6 m (a typical earth radius value) is used for the distance from Earth\'s center, then g will be calculated to be 9.8 m/s^2.

The escape velocity formula is given by:

v_e = (2gR₀)

V_e= (approx 11 km/sec).