What happens when a body is moving in a non-uniform circular motion where tangential acceleration is also present?
Will it continue to move in circular path or it will take an elliptical orbit?
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3$\begingroup$ If it takes an elliptical orbit (with nonzero eccentricity), then it is not, in fact, moving in any circular motion, by definition. $\endgroup$– SandejoCommented Jun 3, 2021 at 19:11
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4 Answers
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What @mia says is true. If you have a rocket orbiting Earth, tangential acceleration that speeds it up will make it spiral away from Earth.
Acceleration the other way will slow it and make it spiral toward the Earth.
If you have a rocket bolted onto a Merry Go Round, where the radius is fixed, the rocket will go in circular motion, faster and faster. Or perhaps slower and slower. In this case, the centripetal acceleration changes to match what is needed to keep motion circular.
answered Jun 3, 2021 at 21:07
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When an object is moving in two or more dimensions, the tangential component of acceleration affects the speed while the perpendicular component changes the direction. If you start with something moving in uniform circular motion and add some tangential acceleration without changing the centripetal acceleration, it will spiral away from the center and never come back.
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It depends on the constraint. E.g. on whether the path is fixed or not, or whether one acceleration is required constant or not.
If we are talking about, say, an orbiting celestial object (or satellite) in space, then the path is not fixed. With a tangential acceleration apart from the centripetal acceleration due to gravity, we will thus see an elliptic path forming.
If we are talking about, say, a car driving around a roundabout, then the driver will keep the car on the fixed path even while applying tangential acceleration (speeding up/down). He does so by adjusting the steering wheel which changes the friction, the centripetal force and thus the centripetal acceleration.
What is impossible, though, is to change the tangential acceleration without changing the centripetal acceleration and still keep a circular path. The tangential acceleration would change the speed $v$, and in order for us to keep a circular path, the blow relationship must be upheld, which requries a changing centripetal acceleration:
$$a_\text{centripetal}=\frac{v^2}{r}.$$
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@mia and @mmesser314 are correct, but I would just add that it all depends on the forces present. The tangential acceleration (caused by whatever tangential forces are present) will result in a larger tangential speed and in order to maintain the same radius you need a larger radial force $F_r = m v^2/r$ If you have a kid on a merry-go-round, then the force of static friction increases as the tangential speed increases until at some point it reaches the maximum $\mu_s N$ and can not provide the necessary centripetal acceleration at that radius. The child slips and moves radially outward.
