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Acceleration is directed towards the center of the circle in a uniform circular motion. Is it same for the non-uniform circular motion?

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If the tangential velocity component is changing then there will be an additional component to the acceleration in the tangential direction. The acceleration is $\vec{a}=\mathrm{d}\vec{v}/\mathrm{d}t$, so if you know the velocity as a function of time and how to take derivatives you can work it out for your particular case. – Michael Brown Jan 19 '13 at 9:59
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Assuming that by non-uniform circular motion you mean moving in a circle but at changing speed, then this does not conserve angular momentum so it cannot happen in a central field. There must be so component of the force tangential to the radius, and therefor some component of the acceleration that is not central.

This happens for satellites orbiting the Earth, and indeed the Moon (as the GRAIL satellites have found) because the Earth, Moon and presumably the vast majority of other bodies are not spherically symmetric so field felt by an object orbiting close to them is not central.

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