Gravitation

[MedPR]

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A space ship experiences centripetal acceleration while orbiting a planet. According to Newton's laws, if the spaceship encounters no resisting force in the course of its circular orbit, what will be the future of its path?

A. orbit in a circle forever
B. gradually spiral inward
C & D are irrelevent.

Answer is A.

Two questions about this. Centripetal acceleration (and centripetal force) is always toward the center of the circle, correct? If there's no force resisting the inward force, why wouldn't the object spiral inward? Second, aren't orbits around planets elliptical?

 

[ljc]

Because the object is moving fast enough that it won't hit Earth. Imagine a cannonball, fired fast enough just above earth's surface so that it remains at the same height, despite falling. As fast as it falls, Earth's surface falls away from it. This is how satellites orbit. They ARE falling, they just keep "missing" the earth, by the same exact distance every second. This is why there is 1 speed possible for each orbit.
V^2=GM/r

Given G is a constant, M is the mass of the earth, there can only be one velocity for each orbit (r). So there is one particular speed at which an object will move fast enough at a given distance from Earth in order to keep "missing." Any slower or faster, and the object will adjust its orbit or else spiral downward.

And yes, I believe orbits are elliptical, but for the purposes of the MCAT, I would venture to guess they should be considered purely spherical.

 

[milski]

It's pretty much the definition of circular motion - a constant force which is always perpendicular to the velocity. The force does turn the velocity vector slightly towards the center of the planet but by the time that happens, the space ship has moved a bit further ahead and now the centripetal force is again perpendicular to it, trying to move it in a slightly different direction. Spiraling inward would mean that this force is not pointed straight towards the center of the earth but slightly backwards so that it slows down the ship.

Another way to think about is to consider the ship's energy. If the force is always perpendicular to the velocity, there won't be any changes to the size of the velocity vector. In other words, the speed will stay the same. That means that the KE of the ship will stay the same. But if the force is always perpendicular to its path, it's also not doing any work on the ship. So the total energy of the ship stays the same. E=PE+KE. E and KE don't change, that means PE does not change, that means that the ship stays at a constant distance from the planet.

Yes, they are. But the gravitational force that they experience is not strictly perpendicular to their velocity which causes them to continuously accelerate and slow down. For each speed, you have a single 'stable' circular orbit. If you start at the same speed further away, you'll start getting closer to the planet and accelerating until you are moving so fast that you are increasing the distance from the planet. With no friction, this oscillations will continue forever.

 

[MedPR]

Because the object is moving fast enough that it won't hit Earth. Imagine a cannonball, fired fast enough just above earth's surface so that it remains at the same height, despite falling. As fast as it falls, Earth's surface falls away from it. This is how satellites orbit. They ARE falling, they just keep "missing" the earth, by the same exact distance every second. This is why there is 1 speed possible for each orbit.
V^2=GM/r

Given G is a constant, M is the mass of the earth, there can only be one velocity for each orbit (r). So there is one particular speed at which an object will move fast enough at a given distance from Earth in order to keep "missing." Any slower or faster, and the object will adjust its orbit or else spiral downward.

And yes, I believe orbits are elliptical, but for the purposes of the MCAT, I would venture to guess they should be considered purely spherical.

It's pretty much the definition of circular motion - a constant force which is always perpendicular to the velocity. The force does turn the velocity vector slightly towards the center of the planet but by the time that happens, the space ship has moved a bit further ahead and now the centripetal force is again perpendicular to it, trying to move it in a slightly different direction. Spiraling inward would mean that this force is not pointed straight towards the center of the earth but slightly backwards so that it slows down the ship.

Another way to think about is to consider the ship's energy. If the force is always perpendicular to the velocity, there won't be any changes to the size of the velocity vector. In other words, the speed will stay the same. That means that the KE of the ship will stay the same. But if the force is always perpendicular to its path, it's also not doing any work on the ship. So the total energy of the ship stays the same. E=PE+KE. E and KE don't change, that means PE does not change, that means that the ship stays at a constant distance from the planet.

Yes, they are. But the gravitational force that they experience is not strictly perpendicular to their velocity which causes them to continuously accelerate and slow down. For each speed, you have a single 'stable' circular orbit. If you start at the same speed further away, you'll start getting closer to the planet and accelerating until you are moving so fast that you are increasing the distance from the planet. With no friction, this oscillations will continue forever.

Thank you both. My physics intuition/understanding is slowly getting better thanks in large part to SDN.