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Problem 114 Hard Difficulty

Three uniform spheres are located at the corners of an equilateral triangle. Each side of the triangle has a length of 1.20 $\mathrm{m}$ . Two of the spheres have a mass of 2.80 $\mathrm{kg}$ cach. 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?

Answer

$2.25 \times 10^{-10} \mathrm{m} / \mathrm{s}^{2}$

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Video Transcript

to answer this question, we have to calculate what is the net force acting on the third sphere for them, calculating its acceleration. For that we have to use Newton's second law and I'm using the falling reference frame. Is this white? And these is X. They are my rented access. Then applying Newton's second law two Sphere number tree, we get the following in the X direction. The net force is given by the mass number three times the acceleration off that sphere in the X direction. But the net force that acts on this fear number three in the X direction is composed by true forces because the system has a following forces. So sphere number one is exerting a gravitational force on sphere number Tree like this. Let me call this F one At the same time, Sphere number choo is exerting your force on sphere number three Like this, I noticed that the distance between this fear number two on sphere number three on sphere number one and number three is the same 1.2 meters. The reform the forest that this your number to exert stones fear number three on sweeter number one exerts on speaker number tree have the same magnitude. So these is F one truth then the net force that acts on the X direction is composed by the X component off both these forces. We have the X component off these force and they have the X component off these force. Then then that force is given by F one component X minus F one component X and these is the mask off the third sphere times its acceleration. You can see that because of the symmetry, there is a cancellation in this forces on these results in an acceleration in the X direction that is zero. Now we repeat these to the Y direction, the y direction. We have true forces acting the Y component off the force F one and again the y component off the force F one. Then in the Y direction the net force, which is given by the mass off sphere number three times. It's why acceleration results is in two times f one component quiet being equal to the mass off spear number three times it's worth consideration. Notice that these components points to the negative why their election? So we have to include a minus. Sign here then and why Acceleration is given by minus two times F. One. Why Divided by M Tree You complete F one white. We have to get a relation between F one and F one. Why for that? We use this triangle, which is formed by F one and its components. The Y component and the X component. Notice that this angle is 60 degrees. Then this order angle is an angle off 30 degrees. So these is 30 degrees. It means that this angle it's 30 degrees. So we have a 30 degrees angle here. Then the white component is given by the pair. Sign off 30 degrees, so the kill sign off 30 degrees is af one. Why divided by F one and then f one component. Why is f one times they co sign off. 30. Sharing the Y acceleration is given by miners. Truth times f One times they come sign off 30 degrees, divided by three. But F orm is a gravitational force and we know how to calculate reputational forces. F one is given by the mass off sphere number one or number. True, they have the same masses times the mask off spear. No directory times, new terms constant, divided by the distance. Let me call it el between spheres one entry squared, then the acceleration in the wider action is minus true times and one times M three times g times. They co sign off 30 degrees, divided by M tree times l squared, then the mass entry. Simplify it and we're left with an expression for the acceleration in the Y direction. And these expression is the white acceleration is given by minus two times m one times g times. They co sign off 30 degrees divided by l squared before, So she needs to get in America. Why you let me organize my board And now we can plugging the values to get in America while you Ford acceleration minus truth climbs 2.8 times 6.6 to 7 time stand to minors. 11 finds the co sign off 30 degrees divided by one point true squared and these results an acceleration off approximately two 0.25 times. Stand to my understand we from my no sign meters per second squared se as the X component is equals to zero as we have seen, and in that acceleration is true. 2025 times 10 to the minus 10 meters per second squared downwards. So the magnitude is two people 25 times 10 to my understand.