Question

Three uniform spheres are located at the corners of an equilateral triangle. each side of the triangle has a length of 1.20 m. two of the spheres have a mass of 2.80 kg each. the third sphere (mass unknown) is released from rest. considering only the gravitational forces that the spheres exert on each other, what is the magnitude and direction of the initial acceleration of the third sphere?

Answer 1

Refer to the diagram shown below.

Because of symmetry, equal forces, F, exist between the sphere of mass m and each of the other two spheres.
The acceleration of the sphere with mass m will be vertical as shown.

The gravitational constant is G = 6.67408 x 10⁻¹¹ m³/(kg-s²)

Calculate F.
F = [ (6.67408 x 10⁻¹¹ m³/(kg-s²))*(m kg)*(2.8 kg)]/(1.2 m)²
   = 1.2977 x 10⁻¹⁰ m   N

The resultant force acting on mass m is
2Fcos(30°) = 2*(1.2977 x 10⁻¹⁰m N)*cos(30°) = 2.2477 x 10⁻¹⁰m  N

If the initial acceleration of mass m is a m/s², then
(m kg)(a m/s²) = (2.2477 x 10⁻¹⁰m N)
a = 2.2477 x 10⁻¹⁰ m/s²

Answer:
The magnitude of the acceleration on mass m is 2.25 x 10⁻¹⁰ m/s².
The direction of the acceleration is on a line that joins mass m to the midpoint of the line joining the known masses.