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Two objects, object A with a mass of 55 kilograms and object B with a mass of 12 kilograms, are near an even larger object with a mass of 10 to the 22 kilograms.
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Object A and object B are at an equal distance, 1,000 kilometers, away from the center of mass of the very large object.
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Which of the objects A and B will have the greater acceleration towards the very large object?
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Looking at the diagram, we have two small spherical objects some distance away from a large spherical object.
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We can see that one of these objects has a mass of 55 kilograms.
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So we know that this is object A.
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And the second one has a mass of 12 kilograms, so this must be object B.
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The very large object is shown as a rectangle but labeled as spherical.
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So we know it must be very, very large on this scale, too large to show here.
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So weβre just seeing a small portion of it.
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Weβre given its mass of 10 to the 22 kilograms.
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And weβre also told that the smaller objects A and B are both the same distance away from the very large object at 1,000 kilometers.
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Now that distance is measured to the center of mass of the very large object.
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So the dotted line here indicates that this is actually measured to a point off the screen.
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We know that both objects A and B experience acceleration towards the center of mass of the very large object because of the force of gravity.
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And weβre asked to determine which one of those experiences the greater acceleration.
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So we need to recall the equation for acceleration due to gravity, which is π is equal to πΊπ over π squared, where π is the acceleration due to gravity.
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πΊ is the universal gravitational constant equal to 6.67 times 10 to the minus 11 meters cubed per kilogram second squared.
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π is the mass of the object weβre accelerating towards.
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And π is the distance between the centers of mass of the object experiencing the acceleration and the object itβs accelerating towards.
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So letβs first consider the acceleration of object A, which weβll call π subscript π΄.
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This is equal to πΊπ over π squared, where πΊ is a constant.
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π is the mass of the large object, which is 10 to the 22 kilograms.
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And π is the distance between the center of mass of object A and the center of mass of the large spherical object, which is 1,000 kilometers.
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Now, letβs compare that to the acceleration of object B, which weβll call π subscript B.
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This is also equal to πΊπ over π squared, where πΊ is the same constant, π is the same mass of 10 to the 22 kilograms, and π is the same distance, 1,000 kilometers.
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These are identical, so we can write π subscript A is equal to π subscript B.
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And this is true because both objects are the same distance away from the center of the large spherical object and because the masses of objects A and B do not appear in this equation.
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That is, acceleration due to gravity is independent of the mass of the object experiencing the acceleration.
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Therefore, the answer to the question βWhich of the objects A and B will have the greater acceleration towards the very large object?β is that objects A and B have the same acceleration.