# Is rotational motion conditioned to a central force?

We know rotational motion as a combination (a resultant) of two effects the tangential velocity and a centripetal force. Does rotational motion turn into linear motion at the same instance this centripetal force goes absent, and that rotational motion can't be a natural motion caused by one single effect, and linear motion is the only (raw) motion present in the universe?

We take an example, a spacecraft going to the moon (or anywhere into outer space). That spacecraft will leave the earth's atmosphere, and continue to move in a spiral motion outwards, increasing in radius, due to the momentum acquired by the earth's rotation.

At some point that spacecraft should lose (if I'm correct) this rotational motion and shoot into a straight line, tangent to the last circle whose radius the distance from the earth's center. That's because the gravity (centripetal force) is no longer an acting force. Can this be true??

There's also a scene from the movie "Gravity" I didn't quite absorb. The astronaut keeps moving around when they separate from a rotating object and continue to move that way, (very famous scene in the trailer), is this accurate?

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No tangential force is needed. The centripetal force is enough to keep an object going around. – Brandon Enright Feb 8 '14 at 0:00
@BrandonEnright I edited my question, to best express my case. I meant "tangential velocity". – Ray Feb 8 '14 at 0:13

## 2 Answers

The only force in acting on bodies in circular motion is the centripetal force, equal to $$F=\frac{mv^2}{r}$$ The centripetal force acts to the centre of the rotating body; there is no such thing as a tangential force. Also, gravity acts over infinite range, so the spacecraft will always have the Earth's gravitational field acting on it. If however, the centripetal force is abruptly removed (such as when a string attached to a tennis ball breaks when turned around), the body will follow the tangent to the circle at the moment the force was removed.

As for the scene in Gravity, this was correct, because there is no friction (air resistance) to slow down, or stop the rotation.

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By tangential force I meant the velocity. I will edit that in the question, however the centripetal force is a linear force, isn't it? What is the centripetal force in Gravity, once the tension from the object that was rotating (I also question this object's motion) is removed (the astronaut gets separated from it) and the astronaut continues to move in a rotational motion? – Ray Feb 8 '14 at 0:08
First of all you should define your terms correctly: tangential momentum is not velocity. Also all forces are linear. The scene were the astronaut is separated from the other one is incorrect and just used to add drama to the movie. Both objects (astronaut and spaceship) have the same velocity, so there is no way that there can be tension in the string. – Ruben Feb 8 '14 at 0:17
Ok, but you agree with me that rotational motion is conditional to a central force, and no object in space can move in a rotational pattern on it's own momentum without this force? or "A force" – Ray Feb 8 '14 at 0:28
Yes, that is correct: without external forces the object will move in a straight line (if it had momentum to start with). – Ruben Feb 8 '14 at 0:32

As the spacecraft leaves the Earth, it will immediately go into a straight motion if no force is acting on it. Its linear momentum may be composed of linear momentum inherited from Earth rotation (imparting a velocity to objects on its surface) and possibly other sources of linear momentum that made it leave the planet. Any angular momentum it may have (inherited from Earth, or otherwise) will only result in the spacecraft rotating around its own center of mass.

However there is at least one force acting on the spacecraft: Earth gravity. Its effect will be to bend the trajectory (assuming it is not going straight up) into an ellipse, a parabola or a hyperbola, depending on the velocity of the spacecraft.

There is no spiral motion.

When you let go of a rotating body, you shoot immediately into a straight line, along a tangent, because the centripetal force is gone. That is correct. However it is a straight line only if no other force is acting, else the trajectory may be modified by this other force.

Regarding the film Gravity, there were a few instances when I wondered about the correctness of the physics. But things go too fast to really analyze the situation, and I do not have it on DVD. What you describe does not seem very physical, but I do not recall it: separated from the rotating body, the astronaute should go in a straight line (as I said above). This straight line is only a local approximation as the astronaut will be on an elliptic orbit because of Earth gravity.

Actually Earth gravity was there from the beginning but could be ignored as a first approximation when analyzing a local event taking place in free fall.

Your statement about rotational motion (or angular momentum) being only a special case of linear motion (linear momentum) is actually wrong. But that might take too long to explain, and I am not sure I would do it adequately.

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Could you elaborate more on your last paragraph. Where did I say that angular momentum being only a special case of linear motion? Please explain. – Ray Feb 8 '14 at 3:13
you said "rotational motion can't be a natural motion caused by one single effect, and linear motion is the only (raw) motion present in the universe?". Maybe I misinterpreted your words ... what did you mean ? – babou Feb 8 '14 at 8:57
Thank you. I meant that rotational motion can't exist without an external force, unlike linear motion which can. An object in space can have a uniform velocity without an external force to govern the motion. – Ray Feb 8 '14 at 9:02
@Yoda As far as I know, this last statement is correct for rotational motion (though not for angular momentum, from what I understand), but your initial statement was stronger. But uniform linear velocity is the same as no velocity at all, as it matters only as a relative motion. However angular velocity exists apparently in an absolute sense, and that is why a bucket of water "knows" it is rotating on itself rather than being still with the universe rotating around it. The same is true of any other kind of acceleration. – babou Feb 8 '14 at 10:32
Could you explain more how a bucket of water knows it's rotating. We on earth thought for quite sometime that the universe was spinning around us, aren't they the same? I'm sorry but I'm having a rough time understanding similar paradoxes. – Ray Feb 8 '14 at 10:44