A common belief among undergraduates is that the gravitational force exerted by a homogeneous sphere of mass M on an extended body of mass m with arbitrary shape is always given by Newton’s law of gravity F = GMm/r_(cc)~2, where r_(cc) is the distance from the center of the sphere to the center of mass (c.m.) of the body. In this note, I introduce the simplest counterexample of a vertical dumbbell to show that, in general, this procedure does not return the correct gravitational force. I also show that not even the center of gravity (c.g.) of the body, determined according to the weighted-average formulae found in textbooks, leads to the correct force. Finally, I present an equation for the c.g. position whose solution always corresponds to the correct force.
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机译：本科生普遍认为，质量均匀的球体 M 对具有任意形状的质量 m 的扩展物体施加的引力总是由牛顿万有引力定律 F = GMm/r_（cc）~2 给出，其中 r_（cc） 是从球体中心到物体质心 （c.m.） 的距离。在本笔记中，我介绍了垂直哑铃的最简单反例，以表明，一般来说，这个过程不会返回正确的重力。我还表明，即使是根据教科书中的加权平均公式确定的身体重心（c.g.）也无法导致正确的力。最后，我提出了一个 c.g. 位置的方程，其解总是对应于正确的力。
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