Help with orbital velocity and centripetal force

[Katt7777]

I was hoping someone well versed in math and physics could help me out with a few questions.

Is it possible in theory to orbit the earth and/or other objects in a way that the centripetal force can maintain 1G?

If so what is the formula needed to figure out the orbital velocity required at a specific altitude with gravitational effects (ie.from the planet etc.) being factored in.

In your opinion would this be feasible in the near future?

Thanks

 

[Izack]

You can attain that easily enough: stand still on Earth's surface.

The force of Earth's gravity (which is the centripetal force for orbiting bodies) near sea level will always be equal to what is required to accelerate you at 1G.

 

[francisdrake]

That's easy
I use a simplified approach, assuming that at low orbital height the actual gravity force is still pretty much 1 g.

- If you are in vessel standing still (relatively to Earths center) the vessel had to thrust upward with 1 g to avoid falling down. Anybody inside this vessel would feel their weight at 1 g.

- If you are orbiting in a circular trajectory the centrifugal force and the gravitational force are exactly balanced. The centrifugal force is 1 g, balanced by the gravitation of 1 g. Everybody inside this vessel is weightless.

- To feel an outward force of 1 g, the vessel must fly with higher speed speed and continuously thrust inwards at 1 g. Everybody feels the difference of 2 g centrifugal force minus 1 g gravitational force, resulting in a net outward force of 1 g.

This higher speed is calculated using the formula for the centrifugal force:

a_c = omega² * r = v² / r

Angular speed omega /s
Zentrifugal force a_c 19,62 m/s²
Velocity v 11357 m/s²
Orbital radius r 6571000 m

The zentrifugal force 19,62 m/s² is the double of 1 g.
The 'orbital overspeed' is roughly the escape velocity.

 

[Katt7777]

Thank you for the explanation and formula.:tiphat:

Correct me if I'm wrong but are you saying that a centripetal force of +0G would cause you to leave orbit?

If so it wouldn't be very feasible, since you would be using tons of fuel up to maintain a circular coarse.:facepalm:

 

[River Crab]

Edit: Fail :facepalm:

If you had no tangential velocity, you would have to be hovering on a point to keep your altitude as the Earth rotates beneath you, which is not a circular trajectory at all. You'd be using all your thrust to overcome gravity drag...

But the only way you can control the centripetal force from the parent body is by varying your distance, assuming your spacecraft has an almost-negligible mass compared to the parent...

 

[Izack]

Thank you for the explanation and formula.:tiphat:

Correct me if I'm wrong but are you saying that a centripetal force of +0G would cause you to leave orbit?

If so it wouldn't be very feasible, since you would be using tons of fuel up to maintain a circular coarse.:facepalm:

If you have no centripetal acceleration, you aren't orbiting anything at all. Flying in circles in the middle of nowhere is a bit of a waste of fuel. :lol:

 

[Tommy]

I think you don't quite understand the relationship between centripetal force, centrifugal force, gravity, and inertia.

Newton defined "A centripetal force is that by which bodies are drawn or impelled, or in any way tend, towards a point as to a center." In this case, the centripetal force is supplied by gravity.

A centrifugal force is the opposite - away from the center. It is supplied by inertia (provided by the mass and velocity of the object).

In a circular orbit, centripetal force is balanced by centrifugal force. If you increase the centrifugal force (by increasing the inertia by increasing the velocity) an object will move away from the center - gain altitude.

If you lower the centripetal force (by gaining altitude and reducing the strength of the gravity) you need less centrifugal force to balance - so the higher orbit requires less velocity.

I had the impression you were thinking along the lines of creating an "artificial" gravity on a vessel by orbiting at the right speed. Not possible. The two forces (centripetal and centrifugal) affect the passengers just the same as the vessel - so there is no "relative" gravity. You would need to apply a force to either the passengers OR the vessel (but NOT both) in order to create artificial gravity. You would either need the vessel to be tethered to the planet or would need to use thrust to balance the forces on the vessel.

 

[Fizyk]

In a circular orbit, centripetal force is balanced by centrifugal force.

Uh, oh. That's a common misconception, resulting from mixing of reference frames. Let me explain.

We can look at the orbital movement (let's say it's a circular orbit around Earth) from two frames of reference. The first one is the Earth's frame, the second one is the rocket's frame.

1. Earth's frame: Here we have an orbiting rocket and Earth is standing still. Gravity acts as the centripetal force and it's causing the trajectory of the rocket to curve and form a circle. There is no centrifugal force here, it's an inertial frame of reference.

2. Rocket's frame: Here the rocket is standing still and the Earth is moving. The gravity is still acting on the rocket, but the rocket isn't moving anyway. From Newton's first law we deduce that there must be another force, balancing the force of gravity. It's the centrifugal force. It's a non-inertial frame of reference.

Why can't we say that the centripetal force is balanced by the centrifugal force? Well, when the forces are balanced, there is no place for moving in circles - Newton's first law again, when forces are balanced, body remains at rest or moves in a straight line.

I hope it's clearer now

 

[tblaxland]

1. Earth's frame: Here we have an orbiting rocket and Earth is standing still. Gravity acts as the centripetal force and it's causing the trajectory of the rocket to curve and form a circle. There is no centrifugal force here, it's an inertial frame of reference.

[Pedantry]If we stick to the world according to Newton, it's an approximately inertial frame of reference. The Earth is still going around the Sun [/Pedantry]

Why can't we say that the centripetal force is balanced by the centrifugal force? Well, when the forces are balanced, there is no place for moving in circles - Newton's first law again, when forces are balanced, body remains at rest or moves in a straight line.

It is much easier if you consider that the orbiting body is following an inertial path in curved space. Then both pseudo-forces (gravity and centrifugal force) disappear.

 

[Fizyk]

[Pedantry]If we stick to the world according to Newton, it's an approximately inertial frame of reference. The Earth is still going around the Sun [/Pedantry]


It is much easier if you consider that the orbiting body is following an inertial path in curved space. Then both pseudo-forces (gravity and centrifugal force) disappear.

True and true. I just didn't want to go into too much detail

Also, while the idea of curved space might be easier, the calculations usually are not

 

[Tommy]

Uh, oh. That's a common misconception, resulting from mixing of reference frames. Let me explain.

We can look at the orbital movement (let's say it's a circular orbit around Earth) from two frames of reference. The first one is the Earth's frame, the second one is the rocket's frame.

1. Earth's frame: Here we have an orbiting rocket and Earth is standing still. Gravity acts as the centripetal force and it's causing the trajectory of the rocket to curve and form a circle. There is no centrifugal force here, it's an inertial frame of reference.

Reference frames are an intellectual crutch. The exist only in our minds - not in the natural universe. They help us understand things - but also limit our understanding.

Let's take your Earth frame example. You say that gravity provides a centripital force. By definition, that means a force that wants to bring the objects toward the center. Yet the rocket isn't moving closer to the center - it's maintaining it's distance. This means that one of two things is occurring.

1. There is a force opposing the centripital force. Since centripital force causes movement TOWARD the center, an opposing force will cause a movement AWAY from the center - and that force meets the description of Centrifugal force. If the rocket is in a circular orbit, niether moving to the center, or away from it, those forces MUST be "balanced".

2. There is no force acting on the vessel that would bring it to the center or away from the center. This would mean that gravity is NOT affecting the rocket.

Of course, both centripetal and centrifugal forces are also human inventions (as tblaxland said, psuedo-forces), just like reference frames!

 

[Fizyk]

Reference frames are an intellectual crutch. The exist only in our minds - not in the natural universe. They help us understand things - but also limit our understanding.

Even so, you still need a reference frame to describe anything. There is no way of telling where the rocket is and how is it moving, if you don't have any object to relate to (aka, a reference frame).

Let's take your Earth frame example. You say that gravity provides a centripital force. By definition, that means a force that wants to bring the objects toward the center. Yet the rocket isn't moving closer to the center - it's maintaining it's distance. This means that one of two things is occurring.

1. There is a force opposing the centripital force. Since centripital force causes movement TOWARD the center, an opposing force will cause a movement AWAY from the center - and that force meets the description of Centrifugal force. If the rocket is in a circular orbit, niether moving to the center, or away from it, those forces MUST be "balanced".

2. There is no force acting on the vessel that would bring it to the center or away from the center. This would mean that gravity is NOT affecting the rocket.

I think we'll all agree that 2 is absurd, but 1 is wrong too. If the forces were balanced, the rocket would be moving along a straight line. It obviously doesn't do that, it moves along a circle (or an ellipse). This means there is an unbalanced force, which is the centripetal force, which is gravity.

The whole point of an orbit is that the rocket is constantly falling. It is trying to move closer to the center because of unbalanced force of gravity, but its tangential velocity is so high, that before it falls, it already misses the Earth.

Of course, both centripetal and centrifugal forces are also human inventions (as tblaxland said, psuedo-forces), just like reference frames!

If you want to treat both gravity and centrifugal force as pseudo forces, then the description of the movement of the rocket looks like this: no forces act on the rocket, so its world line is a generalization of a straight line, called a geodesic. The Earth causes the spacetime to curve, so the geodesics look like helices and in effect the rocket orbits the Earth.

 

[Tommy]

If the forces were balanced, the rocket would be moving along a straight line.

Not true - unless the vessel has no velocity relative to the body it is orbiting (in a local frame - not counting the movement of the planet). Given any relative velocity - and the vessel not moving (or accelerating) toward or away from the center, the result will be a circle.

Centrifugal force isn't defined as a force that makes things tend to move in a straight line. It is defined as a force "away from the center". Inertia makes an object tend to travel in a staight line, and the part of that force that is away from the center meets the definition of Centrifugal force. Remember - the assignment of forces as being Centripetal or Centrifugal is arbitrary, a "convention" used to ease communication - but not "real". A single force can meet more than one definition at a time - and meet different definitions under differing circumstances.

Even so, you still need a reference frame to describe anything. There is no way of telling where the rocket is and how is it moving, if you don't have any object to relate to (aka, a reference frame).

Whether or not we humans have defined and measured something doesn't mean it doesn't exist - and whenever you define something you place artificial limits on the understanding of that object or idea. Since humans aren't omnipotent, we need to define things so that we can limit our ideas to a "size that fits in our heads". But that definition DOES NOT determine reality - only our limited perception of it.

That's why so many people can't comprehend quantum theory - in order for it to work things need to be left undefined and unobseved while we work with them, and can only be defined when the desired results have been achieved. We have an almost instictive need to stick a fork in the cake to see if it's done - but in quantum mechanics forking the cake will prevent the cake from ever being done - may, in fact, make the cake have never existed in the first place, or make it a burrito.

If you want to treat both gravity and centrifugal force as pseudo forces, then the description of the movement of the rocket looks like this: no forces act on the rocket, so its world line is a generalization of a straight line, called a geodesic. The Earth causes the spacetime to curve, so the geodesics look like helices and in effect the rocket orbits the Earth.

This is less "limiting" than the Newtonian perspective, still a bit arbitrary - but likely closer to the "real truth". It's also harder to understand and work with - which is why arbitrary forces like Centripetal and Centrifugal are used. They are simpler concepts that work "well enough".

 

[Fizyk]

Not true - unless the vessel has no velocity relative to the body it is orbiting (in a local frame - not counting the movement of the planet). Given any relative velocity - and the vessel not moving (or accelerating) toward or away from the center, the result will be a circle.

To get a circle, you need to accelerate towards the center. The acceleration needed to get a circle is exactly the centripetal acceleration.

Mathematically, the mistake you're making is confusing [math]\frac{d^2 r}{dt^2}[/math] with the radial component of [math]\frac{d^2 \vec{r}}{dt^2}[/math] ([math]r[/math] being the radial coordinate, [math]\vec{r}[/math] being the position vector).

Actually, in polar coordinate system:
[math]\vec{a} = \frac{d^2 \vec{r}}{dt^2} = \frac{d^2}{dt^2}\left[ \begin{array}{c}
r \\
\phi
\end{array}\right] = \left[ \begin{array}{c}
\frac{d^2 r}{dt^2} - r \left( \frac{d\phi}{dt} \right)^2 \\
r \frac{d^2 \phi}{dt^2} + 2 \frac{dr}{dt} \frac{d\phi}{dt}
\end{array}\right][/math]
(If you don't believe me, I can derive it, but it will involve differential geometry. This can be also found at http://en.wikipedia.org/wiki/Polar_coordinate_system#Vector_calculus).
It's that complicated because it's a curvilinear coordinate system. Even if r is constant (and hence [math]\frac{d^2 r}{dt^2}=0[/math]), the radial component of the acceleration may still be nonzero because of the term [math]-r\left( \frac{d\phi}{dt} \right)^2[/math].

BTW, guess what? [math]\frac{d\phi}{dt}[/math] is the angular velocity [math]\omega[/math], so this term is actually [math]-\omega^2 r[/math] - reminds you of anything?

EDIT:
It's also good to remember when centrifugal force appears. Contrary to what some people seem to think, it's not when a body goes in circles. It's when the frame of reference rotates.

Actually, if you define your frame of reference as a frame, which is located at the center of Earth and is rotating with the same angular velocity that the rocket has, then indeed you can say that the centrifugal force balances the force of gravity. There is no centripetal force then, however, because in that frame of reference the rocket is not moving. It is floating in one place. Forces are balanced, rocket is constantly at rest, everything is fine again.

 

[jthill]

The number of different meanings of "force" and "reference frame" being used in this thread is getting hard to track.

So what's one more viewpoint? Here's mine:

The first thing to notice is that we don't feel the force of gravity directly. What we feel are the electrical signals generated as a result of chemical reactions triggered by our bodies' attempt to counteract differences in the accelerations applied to various parts of our bodies. We feel weight because among other things our muscles burn fuel tensing to counteract the effects induced by the ground pushing up on the skin of our backs as we look at the night sky. Forces are inferred, not felt, and gravity doubly so.

If you put a stone in a sling and whirl it, you're applying centripetal "force" to the stone and you feel "the" equal and opposite centrifugal "force", mostly in your hands. If instead of a stone and sling it's your giggling grandchild attached directly to your hands, it's still the same, only better.

The only reason we call one centrifugal and the other centripetal is because the barycenter is so much closer to one body than the other. (If it's you and your lover, I hope you're both experiencing centripetal force in every sense). The centrifugal force relevant when discussing a satellite's orbit is the force *on the earth* that accelerates it toward the satellite about the combined earth-satellite barycenter. It's the satellite tugging on the earth.

What makes a satellite gain or lose altitude is not centrifugal force.

If you point your deltaglider's nose and flight path straight up, then cut the engines, you still gain altitude for quite a while despite having no force accelerating you upward. You gain altitude because of the radial component of your velocity. The same happens if you're pointing any other non-downward direction when you cut the engines.

I like to call what happens "ballistic lift". The important thing to track is that the apparent radial acceleration that results from the tangential component of your velocity really is an illusion: the surface dropping out from under you is the earth is moving the goalposts, so to speak. The coordinate system itself is distorted, founded on an old, old error.

The flat-earth theory is still as valid as it ever was, and that is approximately "startlingly good". It's easy to forget how unusual the circumstances that show its weaknesses really are, because they're so unusual we habitually forget about them entirely. In day to day life their consequences run the gamut from immeasurable to insignificant, so we live and think as if the earth really were flat. For practically all intents and purposes, it is.

It's one of the reasons I love Orbiter.

 

[Fizyk]

If you put a stone in a sling and whirl it, you're applying centripetal "force" to the stone and you feel "the" equal and opposite centrifugal "force", mostly in your hands.

Ok, but that's a different meaning of the centrifugal force. This one is just a reaction to the centripetal force, and it acts on the body, that exerts the centripetal force on the other body. As you noticed:

The centrifugal force relevant when discussing a satellite's orbit is the force *on the earth* that accelerates it toward the satellite about the combined earth-satellite barycenter. It's the satellite tugging on the earth.

It's true, but as such it can't in any way balance the force of gravity acting on the rocket, as it doesn't even act on the rocket.

The other possible meaning, which we are discussing here, is the pseudo force that appears when the frame of reference is rotating. This one would act on the rocket. As I said in my previous post, there is a frame of reference in which it balances the force of gravity, but to say that it balances the centripetal force is still wrong.

The important thing to track is that the apparent radial acceleration that results from the tangential component of your velocity really is an illusion: the surface dropping out from under you is the earth is moving the goalposts, so to speak.

I'd rather say that the lack of acceleration is an illusion. In a circular orbit you are at a constant altitude, so it may look like you're not accelerating. That's not true. You are constantly being accelerated downwards by the gravity, but as you said, "the Earth is moving the goalposts", so the altitude remains constant despite that.

(Actually, that's in the Newtonian view. In the relativistic view, in a way you indeed aren't accelerating in orbit, but that's a bit more complicated.)

 

[tblaxland]

Not true - unless the vessel has no velocity relative to the body it is orbiting (in a local frame - not counting the movement of the planet). Given any relative velocity - and the vessel not moving (or accelerating) toward or away from the center, the result will be a circle.

You are saying "the vehicle is not accelerating" because it is not moving closer to the central body. Such a statement is false. Consider the orbiting body's velocity vector "v(t)" at time "t=0". Now consider it's velocity vector "v(t1)" sometime later at "t=t1". The magnitude of the velocity vector is still the same (we're talking about circular orbits here) but the direction of the velocity vector is different. The difference between v(0) and v(t1) is the acceleration over that time. The instantaneous acceleration is simply the limit of (v(t1)-v(0))/t1 for t1->0 and is equal to gravity.