Last updated on April 16th, 2021 at 09:42 am
In this post, we will discuss and derive the formula for acceleration due to gravity at a depth h. As the depth from the earth’s surface is increased, the value of g falls. We will discuss and derive the equation that describes the change in the value of g due to depth from the earth’s surface.
The formula for acceleration due to gravity at depth h
The formula for acceleration due to gravity at depth h is expressed by the formula:
g2 = g (1 – h/R).
Here g2 is the acceleration due to gravity at depth h and R is the radius of the earth.
g denotes acceleration due to gravity on the earth’s surface.
For example, considering g = 9.8 m/s^2 on the earth’s surface, g2 at a depth of 1000 meter from the surface of the earth becomes 9.7984 m/s^2. [ check with online calculator ]
Derive the Formula for acceleration due to gravity at depth h
Let’s say, a body of mass m is resting at point A, where A is at a depth of h from the earth’s surface.
The distance of point A from the center of the earth = R – h,
where R is the radius of the earth.
Mass of inner sphere = (4/3). Π. (R-h)3. ρ
Here ρ is the density. and Π is 22/7.
Now at point A, the gravitational force on the object of mass m is
F = G M m/ (R-h)2
= G. [(4/3). Π. (R-h)3. ρ] m/(R-h)2
= G. (4/3). Π. (R-h). ρ. m
Again at point A, the acceleration due to gravity (say g2) = F/m = G. (4/3). Π. (R-h). ρ _________________ (7)
Now we know at earth’s surface, g = (4/3) Π R ρ G ( see the proof here: g formula on the surface of the earth using density)
Taking the ratio, again,
g2/g
= [G. (4/3). Π. (R-h). ρ ] / [(4/3) Π R ρ G]
= (R-h) / R = 1 – h/R.
=> g2 = g (1 – h/R)
The formula for acceleration due to gravity at depth h is represented with this equation:
=> g2 = g (1 – h/R) ______ (8)
g1 is acceleration due to gravity at a depth h
Use our online calculator to test the equation. Click the image below for the calculator page.
