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this is a question I had when I was a kid. I'm a bit ashamed because I think I am missing out on something very obvious since I have the same question despite almost being an engineer now!

From Newtonian physics, I understand how although the gravitational force pulls the planet towards a star, and the planet 'falls' towards the star, due the angular momentum, it also moves laterally. In case of a planetary orbit, this is just enough to keep it moving around it in an elliptical orbit. (With speed and radius being such that angular momentum and energy is conserved)

But let's consider a single planet and star system where the planet moves around the star in a circular path for simplicity. If gravitational force provides centripetal force, what can account for the centrifugal force?

I have a feeling it is related to some inaccurate view I have, of centrifugal force. The xkcd comic and wikipedia article talk about two concepts: fictitious and reactive centrifugal force. But it would be safe to assume the star(EDIT: sorry, I meant to choose a frame of reference such that the centripetal force on the planet has to be cancelled by a centrifugal force on the planet in the direction: star to planet) as a stationary frame of reference for the purpose of this question, right? Which means that it is not a fictitious force, right? Or is it not considered a centrifugal force if you take the star as the reference frame?

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Perhaps you are thinking of the centrifugal force as something that prevents the planet from falling into the star? There is only centrifugal force in the orbiting frame of the planet. In this frame, the planet is not accelerating, so you a need centrifugal force to balance the centripetal force. It is perfectly valid to consider the star as your reference frame, but then there is no centrifugal force at all, and the planet is accelerating towards the star.

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Newton's third law tells us that action and reaction are equal and opposite, so the gravitational force between e.g. the Sun and the Earth pulls equally on both bodies:

CentralForces

The Sun moves a lot less, but that's because it's a lot heavier than the Earth. In fact both the Sun and the Earth are accelerating towards their mutual centre of mass so there aren't inwards and outwards forces - both forces are inwards towards the centre of mass. Anyhow, we call the (apparently) outward force on the Sun the centrifugal force from the Latin for fleeing the centre. The (apparently) inwards force on the Earth we call the centripetal force from the Latin for seeking the centre.

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I would think that centrifugal force is provided by inertia. Inertia is usually what will resist acceleration, when you apply a force to a body that can move freely (but my knowledge of physics is pretty old, and for all I know, it may have changed :)

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The previous answers are well and good. Except I find it a bit strange, nonetheless, saying that the moon does not fall to Earth because its inward centripetal acceleration is balanced by its outward centrifugal acceleration. More correct might be to say that the inward gravitational force is balanced by the outward inertial or centrifugal force. Or that the centripetal force caused by gravity is balanced by the centrifugal force. Thus gravity plays the same role as the tension in the string holding a stone as we swing it round our heads. It is interesting to think of the centrifugal force as real one. Astronauts in a centrifuge feel it pressing them outward. And the fact that these devices are called centrifuges implies that in certain cases it is is indeed more convenient to consider this fictitious force as in some sense real.

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