Ch0117 - Chapter 17 The Universal Law of Gravitation 17 The...

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Chapter 17 The Universal Law of Gravitation 104 17 The Universal Law of Gravitation Consider an object released from rest an entire moon’s diameter above the surface of the moon. Suppose you are asked to calculate the speed with which the object hits the moon . This problem typifies the kind of problem in which students use the universal law of gravitation to get the force exerted on the object by the gravitational field of the moon, and then mistakenly use one or more of the constant acceleration equations to get the final velocity. The problem is: the acceleration is not constant. The closer the object gets to the moon, the greater the gravitational force, and hence, the greater the acceleration of the object. The mistake lies not in using Newton’s second law to determine the acceleration of the object at a particular point in space; the mistake lies in using that one value of acceleration, good for one object-to-moon distance, as if it were valid on the entire path of the object. The way to go on a problem like this, is to use conservation of energy. Back in chapter 12, where we discussed the near-surface gravitational field of the earth, we talked about the fact that any object that has mass creates an invisible force-per-mass field in the region of space around it. We called it a gravitational field. Here we talk about it in more detail. Recall that when we say that an object causes a gravitational field to exist, we mean that it creates an invisible force-per-mass vector at every point in the region of space around itself. The effect of the gravitational field on any particle (call it the victim) that finds itself in the region of space where the gravitational field exists, is to exert a force, on the victim, equal to the force-per- mass vector at the victim’s location, times the mass of the victim. Now we provide a quantitative discussion of the gravitational field. ( Quantitative means, involving formulas, calculations, and perhaps numbers. Contrast this with qualitative which means descriptive/conceptual.) We start with the idealized notion of a point particle of matter. Being matter, it has mass. Since it has mass it has a gravitational field in the region around it. The direction of a particle’s gravitational field at point P , a distance r away from the particle, is toward the particle and the magnitude of the gravitational field is given by 2 m G = g (17-1) where: G is the universal gravitational constant: 2 2 11 kg m N 10 67 6 × = . G m is the mass of the particle, and is the distance that point P is from the particle. In that point P can be any empty (or occupied) point in space whatsoever, this formula gives the magnitude of the gravitational field of the particle at all points in space. Equation 17-1 is the equation form of Newton’s Universal Law of Gravitation 1 . 1
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Ch0117 - Chapter 17 The Universal Law of Gravitation 17 The...

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