Spherical particles (proof)

[Demonsthenes]

Homework Statement

Three identical spherical particles of mass m are each located at the corners of an equilateral triangle with edge length r (each sphere is centered at each corner). Show the magnitude of the net gravitational force on anyone of the particles due to the other two particles is given by:

Homework Equations

Fg = sqrt(3) Gm^2/r^2

The Attempt at a Solution

Again... had no idea where to start.

 

[huyen_vyvy]

what's the gravitational force between each pair? and then sum the 2 vectors (i.e, 2 forces acting on 1 particle) using geometry to find the magnetude of the force.

 

[m2003]

I can understand that this problem requires knowledge of Newton's Law of Universal Gravitation. This law states that the magnitude of the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. In this case, we have three identical spherical particles with mass m, located at the corners of an equilateral triangle with edge length r.

To find the net gravitational force on one of the particles, we can use the concept of vector addition. The force on one particle due to the other two can be broken down into two components - one along the line connecting the two particles and one perpendicular to it. Since the triangle is equilateral, the angle between the two particles is 60 degrees. Using trigonometry, we can find that the perpendicular component of the force is equal to (sqrt(3)/2) times the magnitude of the force along the line connecting the particles.

Now, using the Law of Universal Gravitation, we can write the magnitude of the force along the line connecting the particles as Gm^2/r^2. Therefore, the magnitude of the net force on one particle due to the other two is given by:

Fnet = (sqrt(3)/2) * (Gm^2/r^2) + (Gm^2/r^2) = (3/2) * (Gm^2/r^2) = sqrt(3) * (Gm^2/r^2)

This matches the given equation, proving that the magnitude of the net gravitational force on one particle due to the other two is indeed given by sqrt(3) * (Gm^2/r^2).