1. Understanding the Concepts
* Newton's Law of Universal Gravitation: The force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
* F = G * (m1 * m2) / r^2
* F = force of gravity
* G = gravitational constant (6.674 x 10^-11 N m^2/kg^2)
* m1 and m2 = masses of the objects
* r = distance between their centers
* Equilibrium: The particle will experience equal attractions when the gravitational force exerted by the sun is equal to the gravitational force exerted by the Earth.
2. Setting up the Equation
Let:
* `M` be the mass of the Sun
* `m` be the mass of the Earth
* `x` be the distance between the particle and the Sun
* `(1 AU - x)` be the distance between the particle and the Earth (1 AU is the average distance between the Earth and the Sun, approximately 149.6 million kilometers)
We can set up the equation for equilibrium:
```
G * M * m / x^2 = G * m * m / (1 AU - x)^2
```
3. Simplifying the Equation
We can cancel out the gravitational constant (`G`) and the mass of the particle (`m`) on both sides:
```
M / x^2 = m / (1 AU - x)^2
```
4. Solving for x
* Cross-multiply: M(1 AU - x)^2 = m * x^2
* Expand: M(1 AU^2 - 2 * 1 AU * x + x^2) = m * x^2
* Rearrange: (M - m) x^2 - 2 * M * 1 AU * x + M * 1 AU^2 = 0
This is a quadratic equation. You can solve for `x` using the quadratic formula:
```
x = [ -b ± √(b^2 - 4ac) ] / 2a
```
Where:
* a = (M - m)
* b = -2 * M * 1 AU
* c = M * 1 AU^2
5. Finding the Solution
Plug in the values for the mass of the Sun (M = 1.989 × 10^30 kg), the mass of the Earth (m = 5.972 × 10^24 kg), and 1 AU (149.6 million km) to solve for `x`. You will get two solutions, but only one will be physically meaningful (within the Earth-Sun system).
Important Note: The solution will be a distance in astronomical units (AU). You can convert it to kilometers or other units as needed.
Let me know if you'd like to see the complete numerical solution!
