Prob Regarding Gravitational Force

[EL ALEM]

Homework Statement

Three uniform spheres are located at the corners of an equilateral triangle. each side of the triangle has a length of 1.2m. two of the spheres have a mass of 2.8kg 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?

Homework Equations

F=G(m₁m₂/r²)

The Attempt at a Solution

I oriented the triangle so the two spheres with a known mass are along the y-axis, therefore the x-component of the forces cancel via symmetry and we're left w/ the y-component which is (2Gm₁m₂sin60)/r²
Am i doing this right so far?

 

[anathema]

Yes, you are on the right track. To find the magnitude of the initial acceleration of the third sphere, you can use Newton's second law, F=ma, where F is the net force acting on the third sphere and m is its mass. In this case, the net force is the sum of the gravitational forces exerted by the other two spheres, which you have correctly calculated using the equation F=G(m1m2/r^2). As you mentioned, the x-components of these forces will cancel out, leaving only the y-component. Therefore, you can set up the equation F=ma as follows:

(2Gm1m2sin60)/r^2 = ma

Now, you can solve for the magnitude of the initial acceleration, a, by dividing both sides by the mass of the third sphere, m. This will give you:

a = (2Gm1m2sin60)/(mr^2)

Since you know the values for m1, m2, and r, you can plug them into the equation to find the numerical value for the initial acceleration. Remember to use the appropriate units for mass (kg) and distance (m) to get the correct units for acceleration (m/s^2).

I hope this helps! Let me know if you have any further questions.