To derive an expression for the acceleration due to gravity '
g ' at the surface of the Earth, in terms of its radius 'R', its uniform density '
ρ ', and the gravitational constant '
G ', we can start by calculating the mass '
M ' of the Earth in terms of its density and volume.
The volume '
V ' of a sphere is given by
V=πR3For a sphere with uniform density '
ρ ', the mass '
M ' can be calculated by multiplying the volume by the density:
M=ρV=ρ(πR3) Now that we have the mass, we can use Newton's law of universal gravitation to find the force '
F ' exerted on a mass 'm' at the surface of the Earth:
F=The acceleration '
g ' due to gravity at the Earth's surface is simply the force per unit mass 'm':
squareg==Substituting the expression for 'M' in terms of '
ρ ' and '
R ', we get:
g=When we simplify this expression, we have:
g=g=πGρRTherefore, the correct answer is:
Option C
g=πρGR