Three uniform spheres are located at the corners of an equilateral triangle. Each side of the triangle has a length of 2.40 m. Two of the spheres have a mass of 3.40 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?

I tried using the equation G*m1*m3/r^2 and am not getting the correct answer

Consider

m3 to be at (0,0)
m1 exerts its force at an angle of 30°
m2 exerts its force at an angle of -30°
As you say, each force is F=GMm/r^2

The resultant force will be at 0°, with a magnitude of F√3, the sum of the x-components of the two forces.

To find the magnitude of the initial acceleration of the third sphere, we can use Newton's law of universal gravitation. The equation you mentioned, G * m1 * m2 / r^2, is indeed the correct equation to use, where G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between their centers.

First, let's calculate the distance between the centers of the spheres. Since they form an equilateral triangle, all three sides are equal. The length of each side is given as 2.40 m. Thus, the distance between the centers of the spheres is also 2.40 m.

Now, let's calculate the gravitational force exerted between the third sphere (mass m3) and one of the other spheres (mass m1 or m2). Since the masses of the two spheres are given as 3.40 kg each, we can substitute these values into the equation.

F = (G * m1 * m3) / r^2
F = (6.67 x 10^-11 N m^2 / kg^2) * (3.40 kg) * (3.40 kg) / (2.40 m)^2
F = (6.67 x 10^-11 N m^2 / kg^2) * (3.40 kg)^2 / (2.40 m)^2

Now, let's calculate the gravitational force between the third sphere and the other sphere on the opposite side of the equilateral triangle. The distance between their centers is still 2.40 m, and the masses are the same.

F = (6.67 x 10^-11 N m^2 / kg^2) * (3.40 kg)^2 / (2.40 m)^2

Since the two forces are acting in opposite directions, we need to subtract one force from the other to find the net gravitational force on the third sphere.

F_net = F - F
F_net = 2F

Finally, we can use Newton's second law of motion, F = m * a, to find the magnitude of the initial acceleration (a) of the third sphere.

a = F_net / m3
a = (2F) / m3

Now substitute the value of F_net, and we can find the magnitude of the initial acceleration.