https://www.superphysics.org/research/newton/principia/book-1/sec-08/ Recent content in The Invention of Orbits Wherein Bodies will Revolve on Superphysics Hugo en https://www.superphysics.org/research/newton/principia/book-1/sec-08/prop-40/ Mon, 01 Jan 0001 00:00:00 +0000 https://www.superphysics.org/research/newton/principia/book-1/sec-08/prop-40/ <h2 id="proposition-40-theorem-13-if-a-body-acted-upon-by-any-centripetal-force-is-moved-and-another-body-ascends-or-descends-in-a-right-line-and-their-velocities-be-equal-in-any-one-case-of-equal-altitudes-their-velocities-will-be-also-equal-at-all-equal-altitudes">Proposition 40 Theorem 13: If a body, acted upon by any centripetal force is moved, and another body ascends or descends in a right line, and their velocities be equal in any one case of equal altitudes, their velocities will be also equal at all equal altitudes.</h2> <p>Let a body descend from A through D and E, to the centre C; and let another body move from V in the curve line VIKk. From the centre C, with any distances, describe the concentric circles DI, EK, meeting the right line AC in D and E, and the curve VIK in I and K. Draw IC meeting KE in N, and on IK let fall the perpendicular NT; and let the interval DE or IN between the circumferences of the circles be very small; and imagine the bodies in D and I to have equal velocities. Then because the distances CD and CI are equal, the centripetal forces in D and I will be also equal.</p> https://www.superphysics.org/research/newton/principia/book-1/sec-08/prop-41/ Mon, 01 Jan 0001 00:00:00 +0000 https://www.superphysics.org/research/newton/principia/book-1/sec-08/prop-41/ <h2 id="proposition-41-problem-28-supposing-a-centripetal-force-of-any-kind-and-granting-the-quadratures-of-curvilinear-figures-it-is-required-to-find-as-well-the-trajectories-in-which-bodies-will-move-as-the-times-of-their-motions-in-the-trajectories-found">Proposition 41 Problem 28: Supposing a centripetal force of any kind, and granting the quadratures of curvilinear figures, it is required to find as well the trajectories in which bodies will move, as the times of their motions in the trajectories found.</h2> <p>Let any centripetal force tend to the centre C, and let it be required to find the trajectory VIKk. Let there be given the circle VR, described from the centre C with any interval CV.</p> https://www.superphysics.org/research/newton/principia/book-1/sec-08/prop-42/ Mon, 01 Jan 0001 00:00:00 +0000 https://www.superphysics.org/research/newton/principia/book-1/sec-08/prop-42/ <h2 id="proposition-42-problem-29-the-law-of-centripetal-force-being-given-it-is-required-to-find-the-motion-of-a-body-setting-out-from-a-given-place-with-a-given-velocity-in-the-direction-of-a-given-right-line">PROPOSITION 42. PROBLEM 29: The law of centripetal force being given, it is required to find the motion of a body setting out from a given place, with a given velocity, in the direction of a given right line.</h2> <p>Suppose the same things as in the three preceding propositions; and let the body go off from the place I in the direction of the little line, IK, with the same velocity as another body, by falling with an uniform centripetal force from the place P, may acquire in D; and let this uniform force be to the force with which the body is at first urged in I, as DR to DF.</p>