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Thread: 2 interesting questions

  1. #1 2 interesting questions 
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    Hi,

    I have 2 questions here that I would like to get some opinions on.

    1. If I slowly lower a pendulum (from a string) from infinite distance onto the surface of a rotating spherical atmosphere-less planet, will the pendulum touch the planet’s surface perpendicularly or at an angle to the normal? Will this angle change with the planet’s rotational velocity?

    2. An airplane is flying above a spherical atmosphere-less planet, such that its angular speed is equal to the planet’s rotational speed (ie it’s ‘geostationary’). The plane then drops a bomb. The vertical direction is straightforward – the bomb accelerates constantly till it hits the planet’s surface. In the horizontal direction, the tangential speed is constant throughout its trajectory, meaning that as the bomb gets closer to the surface, the angular velocity increases. This means that the bomb will hit the surface at a position that is angularly forward of the plane. Is this analysis right?


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  3. #2 Re: 2 interesting questions 
    . DrRocket's Avatar
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    Quote Originally Posted by bobby2009
    Hi,

    I have 2 questions here that I would like to get some opinions on.

    1. If I slowly lower a pendulum (from a string) from infinite distance onto the surface of a rotating spherical atmosphere-less planet, will the pendulum touch the planet’s surface perpendicularly or at an angle to the normal? Will this angle change with the planet’s rotational velocity?
    According to Newtonian gravity the answer to your question is "no". But in generala relativity there is a very tiny effect that makes the answer to this question "yes".

    http://en.wikipedia.org/wiki/Frame-dragging

    Quote Originally Posted by bobby2009
    2. An airplane is flying above a spherical atmosphere-less planet, such that its angular speed is equal to the planet’s rotational speed (ie it’s ‘geostationary’). The plane then drops a bomb. The vertical direction is straightforward – the bomb accelerates constantly till it hits the planet’s surface. In the horizontal direction, the tangential speed is constant throughout its trajectory, meaning that as the bomb gets closer to the surface, the angular velocity increases. This means that the bomb will hit the surface at a position that is angularly forward of the plane. Is this analysis right?
    Airplanes don't fly very well above atmophere-less planets, spherical or otherwise. So let's assume that you have some sort of rocket performing the flying duties. It is constantly accelerating under power and therefore there is no a prior relatinship between the speed of the plane and the gravitational field of the planet -- it is not in orbit.

    Also let's forget about the miniscule Lens-Thirring effect and treat this problem with the usual Newtoian gravitational theory.

    If the planet is rotating very rapidly, then the linear speed of the craft will be quite high, and that linear, tangential speed will be imparted to the bomb when it is dropped. With sufficiently high speed the bomb need not even hit the planet. It is possible, for instance that the tangential speed imparted to bomb exceeds the escape velocity associated with the planet and the altitude of the craft.

    The bomb trajector, even if it does hit the planet, could be either ahead of or behind that of the craft in angular terms, depending on the particulars. The gravitational acceleration will towards the center of the planet, while the tangential velocity is normal to the initial vector from the center of the planet to the craft at the time of the drop. For instance if the gravity of the planet is extremely strong, then the trajectory of the bomb will be nearly a straight line radially towards the center of the planet from the drop point, so the plane may "fly ahead" of the bomb. But if the gravitational field is weak then the trajectory of the bomb may actually intersect the trajectory of the plane.


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  4. #3  
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    If say we are working in the ‘usual’ Newtonian regime where the planet is not too massive and its angular velocity is not very high, and the plane velocity does not exceed the escape velocity, am I right to say that the bomb will always drop in front of the plane in angular position?

    Also, what do you mean by ‘But if the gravitational field is weak then the trajectory of the bomb may actually intersect the trajectory of the plane’?
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  5. #4  
    . DrRocket's Avatar
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    Quote Originally Posted by bobby2009
    If say we are working in the ‘usual’ Newtonian regime where the planet is not too massive and its angular velocity is not very high, and the plane velocity does not exceed the escape velocity, am I right to say that the bomb will always drop in front of the plane in angular position?

    Also, what do you mean by ‘But if the gravitational field is weak then the trajectory of the bomb may actually intersect the trajectory of the plane’?
    1) no

    2) I mean exactly what I said. Under those conditions the plan could bomb itself.

    I think you need to learn some mechanics.
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