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Answer :
The net gravitational force on the 32.0 kg object placed midway between the two masses is zero because the forces exerted by the masses are equal and opposite. To find a point where the net force is zero again, one must solve an equation where the gravitational forces from the 195 kg and 495 kg masses on the 32.0 kg object are equal and opposite.
To find the net gravitational force exerted on a 32.0 kg object positioned midway between two objects with masses of 195 kg and 495 kg, which are 0.440 m apart, one must use Newton's law of universal gravitation:
F = Gm1m2 / r2
Since the 32.0 kg object is midway, the distance from each mass will be half of 0.440 m, which is 0.220 m.
The gravitational forces from each mass will be equal in magnitude but opposite in direction. Because of this symmetry, the net force will be zero, and thus, there is no need to calculate the magnitudes as they would cancel each other out.
For part (b), to find a position where the 32.0 kg object experiences zero net force, we set up an equation where the gravitational forces due to each mass (195 kg and 495 kg) are equal and opposite. Let x be the distance from the 495 kg mass. The gravitational force by the 495 kg mass when the object is at distance x is F1 = G*(495 kg)(32.0 kg) / x2, and the gravitational force by the 195 kg mass when the object is at a distance (0.440 m - x) from it is F2 = G*(195 kg)(32.0 kg) / (0.440 m - x)2. Setting F1 = F2 and solving for x gives us the position of the 32.0 kg mass where the net gravity force is zero.
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