1. Understand the Concepts

* Gravitational Force (mg): This is the approximate force due to gravity near the Earth's surface. It assumes a constant gravitational acceleration (g).

* Actual Gravitational Force: This takes into account the inverse square law, where gravitational force decreases with distance from the center of the Earth.

2. Set Up the Equation

We want to find the height (h) where the difference between the approximate and actual gravitational force is 4%. Let:

* *g* be the acceleration due to gravity at the Earth's surface (~9.8 m/s²)

* *G* be the gravitational constant (~6.674 x 10⁻¹¹ N m²/kg²)

* *M* be the mass of the Earth (~5.972 x 10²⁴ kg)

* *R* be the radius of the Earth (~6.371 x 10⁶ m)

* *m* be the mass of the object

The approximate force is: *F_approx* = *mg*

The actual force is: *F_actual* = *GmM / (R + h)²*

We want: *(F_actual - F_approx) / F_approx* = 0.04

3. Solve for the Height (h)

Substitute the expressions for *F_actual* and *F_approx* into the equation:

[(GmM / (R + h)²) - mg] / mg = 0.04

Simplify:

[GM / (R + h)² - g] / g = 0.04

[GM / (R + h)²] / g = 1.04

GM / (R + h)² = 1.04g

(R + h)² = GM / (1.04g)

R + h = √(GM / (1.04g))

h = √(GM / (1.04g)) - R

4. Calculate the Height

Plug in the values for *G*, *M*, *g*, and *R*. Remember to convert the radius of the Earth to kilometers.

h = √((6.674 x 10⁻¹¹ N m²/kg²) * (5.972 x 10²⁴ kg) / (1.04 * 9.8 m/s²)) - 6.371 x 10⁶ m

h ≈ 3.27 x 10⁶ m ≈ 3270 km

Therefore, there is approximately a 4% difference between the approximate gravitational force and the actual gravitational force at a height of about 3270 kilometers above the Earth's surface.