Three uniform spheres are located at the corners of an equilateral triangle. Each side of the triangle has a length of 1.17 m. Two of the spheres have a mass of 3.97 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 of the initial acceleration of the third sphere?

Resultant acceleration on the third sphere is

(3.352 × 10⁻¹⁰) m/s²

Explanation:

Newton's law states that the gravitational attractive force between two masses

m and M at some distance, d, apart.

F = (GMm/d²)

The diagram of the three masses placed at the edges of an equilateral triangle is presented in the attached image.

The resultant force, Fᵣ = F₁₃ cos 30° + F₂₃ cos 30°

This is only a sum of vertical components of the forces, as the horizontal components of the forces cancel out.

Note that since m₁ = m₂ = m = 3.97 kg,

F₁₃ = F₂₃ = F

Let the mass of the third sphere = M

F = (GMm/d²)

Recall that

The resultant force, Fᵣ = F₁₃ cos 30° + F₂₃ cos 30°

Fᵣ = F cos 30° + F cos 30° = 2F cos 30° = 1.732 F

Net force on the third sphere, Fᵣ = Ma

Fᵣ = 1.732 F = Ma

Recall that F = (GMm/d²)

Ma = 1.732 F = (1.732GMm/d²)

Ma = (1.732GMm/d²)

a = (1.732Gm/d²)

G = (6.674 × 10⁻¹¹) Nm²/kg²

m = 3.97 kg

d = 1.17 m

a = (1.732 × 6.674 × 10⁻¹¹ × 3.97)/(1.17²)

a = (3.352 × 10⁻¹⁰) m/s²

Hope this Helps!!!

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russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.