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The rope example is an example involving a 'pull' force. The centripetal force can also be supplied as a 'push' force, such as in the case where the normal reaction of a wall supplies the centripetal force for a wall of death or a Rotor rider. Newton's idea of a centripetal force corresponds to what is nowadays referred to as a central force.
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Theorem 3 now evaluates the centripetal force in a non-circular orbit, using another geometrical limit argument, involving ratios of vanishingly small line-segments. The demonstration comes down to evaluating the curvature of the orbit as if it were made of infinitesimal arcs, and the centripetal force at any point is evaluated from the speed ...
Unlike the inertial force known as centrifugal force, which exists only in the rotating frame of reference, the reactive force is a real Newtonian force that is observed in any reference frame. The two forces will only have the same magnitude in the special cases where circular motion arises and where the axis of rotation is the origin of the ...
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Next Newton proves his "Theorema II" which shows that if Kepler's second law results, then the force involved must be along the line between the two bodies. In other words, Newton proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.
Look first at one of the two balls. To travel in a circular path, which is not uniform motion with constant velocity, but circular motion at constant speed, requires a force to act on the ball so as to continuously change the direction of its velocity. This force is directed inward, along the direction of the string, and is called a centripetal ...