Derive an expression for acceleration due to gravity at a depth ‘d’ below the earth’s surface.

To derive an expression for the acceleration due to gravity at a depth 'd' below the Earth's surface, we can make use of Newton's law of universal gravitation and the concept of gravitational potential energy.

At any point inside the Earth, the gravitational force experienced by an object can be derived from Newton's law of universal gravitation:

F = G * (m1 * m2) / r²

Where:
F is the gravitational force between two objects,
G is the gravitational constant (approximately 6.67430 × 10^-11 m³ kg^(-1) s^(-2)),
m1 and m2 are the masses of the two objects (in this case, the object and Earth),
and r is the distance between the centers of the two objects (in this case, the distance from the object to the center of the Earth).

Considering a point at depth 'd' below the Earth's surface, the distance 'r' from the object to the center of the Earth can be expressed as:

r = R - d

Where:
R is the radius of the Earth.

Now, let's differentiate the gravitational potential energy (U) with respect to 'd' to obtain the acceleration due to gravity (g) at depth 'd':

dU = -F * dr

Differentiating both sides with respect to 'd':

d²U/dt² = -dF/dr

Since U = -G * (m1 * m2) / r, we can express dU/dr as:

dU/dr = G * (m1 * m2) / r²

Plugging in the expression for 'r' at depth 'd':

dU/dr = G * (m1 * m2) / (R - d)²

Now, we can express the force 'F' as m2 * g, where 'g' is the acceleration due to gravity:

m2 * g = G * (m1 * m2) / (R - d)²

Canceling out 'm2' on both sides:

g = G * m1 / (R - d)²

Finally, we have derived the expression for the acceleration due to gravity at a depth 'd' below the Earth's surface:

g = G * m1 / (R - d)²

where G is the gravitational constant, m1 is the mass of the Earth, R is the radius of the Earth, and d is the depth below the Earth's surface.