Proove that centripetal force = gravitational force.
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When a satellite is in orbit around a planet, gravity is considered to be a centripetal force even though in the case of eccentric orbits, the gravitational force is directed towards the focus, and not towards the instantaneous center of curvature.
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gravity is a centripetal force, and can be envisaged clearly as such in Newtonian mechanics. Centripetal just means a force that is "radially inwards" ("directed towards the centre"). The electric force between two objects of opposite charges, for example, is also clearly centripetal.
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