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Here Newton finds the centripetal force to produce motion in this configuration would be inversely proportional to the square of the radius vector. (Translation: 'Therefore, the centripetal force is reciprocally as L X SP², that is, (reciprocally) in the doubled ratio [i.e., square] of the distance ... .')
This force is directed inward, along the direction of the string, and is called a centripetal force. The other ball has the same requirement, but being on the opposite end of the string, requires a centripetal force of the same size, but opposite in direction. See Figure 2.
The Eötvös effect is the change in measured Earth's gravity caused by the change in centrifugal acceleration resulting from eastbound or westbound velocity.When moving eastbound, the object's angular velocity is increased (in addition to Earth's rotation), and thus the centrifugal force also increases, causing a perceived reduction in gravitational force.
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. [3]: 107
According to this equation, the second force F 2 (r) is obtained by scaling the first force and changing its argument, as well as by adding inverse-square and inverse-cube central forces. For comparison, Newton's theorem of revolving orbits corresponds to the case a = 1 and b = 0 , so that r 1 = r 2 .
All attractive central forces can produce circular orbits, which are naturally closed orbits. The only requirement is that the central force exactly equals the centripetal force, which determines the required angular velocity for a given circular radius. Non-central forces (i.e., those that depend on the angular variables as well as the radius ...
The component of weight force is responsible for the tangential force (when we neglect friction). The centripetal force is due to the change in the direction of velocity. The normal force and weight may also point in the same direction. Both forces can point downwards, yet the object will remain in a circular path without falling down.