Tidal Locking

[MrGamma]

First off... Thank you very much to everyone in this forum who has helped me come to a better understanding of physics.

 

A website has mentioned that eventually the earth will cease to spin and both the earth and the moon will face each other. Tidally locked with one another.

 

What happens after that? Do they collide with each other and form a larger planet? What might the collision ( if that's what would happen ) be equivalent to? Would they merge as one slowly over time? Would they impact with tremendous force and energy and explode? Is there any known observed phenomenon in the universe which demonstrates this on a planetary scale?

[D H]

The Earth and Moon will still be orbiting each other when the Earth becomes tidally locked with the Moon. The Earth will not stop rotating about its axis; it will have a day equal in length to a month -- a future month, that is. Think about it this way: The Moon is tidally locked with the Earth, but it is still spinning about its axis. One lunar "day" is one month long.

[MrGamma]

The Earth and Moon will still be orbiting each other when the Earth becomes tidally locked with the Moon.

 

I currently understand that there are a few theories floating around which differ regarding orbital mechanics.

 

The first theory is the equivalence principle which I do believe is what is used today. Additionally there appears to be a second equivalence principle used in particle physics which is a more precise calculation of gravity which states that mass effects acceleration. My reference...

 

http://science.nasa.gov/headlines/y2004/06may_lunarranging.htm

 

My Question...

 

2 pieces of paper face each other in a vacuum 2miles apart. One is 1 mile square. One is 10miles square.

 

( answer using the original equivalence principal )

 

The two pieces of paper meet at the half way mark at the same time.

 

( answer using the more accurate equivalence principal )

 

The one mile square piece crosses the halfway mark first.

 

 

 

Is this a correct conclusion? Particle Physics currently applies different gravitational laws than what space navigation uses?

 

Assuming what others have told me in the past to be somewhat accurate. Mass is currently not detected as influencing the pull of gravity according to the standard equivalence principal. Would it be a far stretch to think that gravity could be a harmonic phenominon? Meaning the planets orbit due to standing fields rather than mass and inertia?

 

In particular... I am trying to understand the validity of this theory.

 

http://www.mountainman.com.au/news96_p.html

[Sisyphus]

My Question...

 

2 pieces of paper face each other in a vacuum 2miles apart. One is 1 mile square. One is 10miles square.

 

( answer using the original equivalence principal )

 

The two pieces of paper meet at the half way mark at the same time.

 

This is incorrect. It seems like you're misapplying the idea that two different sized objects fall at the same speed. This is true when you have an opposing massive body in common, in that case the Earth. Your example is a different situation. Just remember two things: First, that the gravitational force exerted by each object on the other is exactly equal, and opposite in direction. And second, the equation f=ma. So in your example, you have the same force acting on each paper (the "f" is the same in each equation), but one of them is ten times as massive as the other (the "m" is different). Hence, as the "m" gets larger, the "a" gets smaller. The 1 mile square piece will accelerate ten times as quickly as the 10 mile square piece, and they will meet at their center of gravity, not halfway between them.

 

The same thing happens when you drop a rock off a cliff, just with a more extreme difference. The same force is acting on the rock as on the entire rest of the Earth, but since the Earth is so much more massive than the rock, that force only accelerates the Earth an immeasurably small amount.

[MrGamma]

Understood...

 

Acceleration = (Gravity Force Constant) / Mass of object

 

So... essentially...

 

Object one and Object Two both have their own acceleration applied to the equation?

 

Acceleration One = (Gravity Force Constant) / Mass of 1 mile square paper.

 

Acceleration Two = (Gravity Force Constant) / Mass of 10 mile square paper.

 

And those two Accelerations are applied to determine the point at which they will arrive in the vacuum.

 

But... If Two objects are falling to Earth... There is only one acceleration which is applied?

 

Acceleration = (Gravity Force Constant).

 

Is this the way it works?

[D H]

But... If Two objects are falling to Earth... There is only one acceleration which is applied?

 

Acceleration = (Gravity Force Constant).

 

Is this the way it works?

The equation a=g is an approximation. Newton's law of universal gravitation still applies (hint: the law is universal). The Earth is nearly spherical, with a radius 6378 km and a mass of 5.974*10²⁴ kg. The gravitational force on a 75 kg person standing on the surface of the Earth is [math]F=\frac{GM_{\text{earth}}M_{\text{person}}}{{r_{\text{earth}}}^2}=735\,\text{newtons}[/math]. The acceleration of the person toward the Earth is [math]\frac{735\,\text{newtons}}{75\,\text{kg}} = 9.8\,m/s^2[/math]. The acceleration of the Earth toward the person is [math]\frac{735\,\text{newtons}}{5.974\times10^{24}\,\text{kg}} = 1.23\times10^{-22}\,m/s^2[/math]. This is an immeasurably small value.

 

That 9.8 m/s² is the acceleration at the surface of the Earth. Because the Earth is so large, the acceleration is nearly constant for a good distance above the surface. For example, at 1 kilometer above the surface of the Earth the acceleration is 9.797 m/s².

[MrGamma]

The equation a=g is an approximation.

 

Okay.... It's tough for me to understand Latex at this point... in php...

 

function force($m,$g){
   return $g/$m;
}

$g = 735; // Newtons...
$m = 75; // Persons Weight in KiloGrams
$force = force($m,$g);

echo $force;

 

 

When you come up with...

 

9.8\,m/s^2

 

according to the square it's the rate of acceleration...

 

9.8\,m/s^2 (first second)

19.6\,m/s^2 (second second)

39.2\,m/s^2 (after three seconds)

 

 

My main issue seems to be rooted in the classic equivalence principle. Where it says that mass does not effect acceleration. Objects falling to earth will do so at the same velocity. I imagine this is typically done because the effects are insignificant. However... for the sake of understanding this properly... This mass effecting acceleration is still hypothetical correct?

 

I am referencing this Nasa article.

 

http://science.nasa.gov/headlines/y2004/06may_lunarranging.htm

 

It is my goal to understand orbital mechanics and I am currently under the belief that they do not use the mass of the object to determine acceleration. Which has me wondering... how do they know how to navigate their satellites?

 

What to they do? Just throw on the accelerator and wait for a ping back? If they gave a little bit too much throttle they play it by ear and adjust?

 

Or are they actually using the gravitational force of the planets in their navigation budgets? Are they always using 735 Newtons? If not... how do I figure out the Newtons of the objects... or theoretically... should I always just use 735 in all my equations. If this is a stupid question I apologize.

Edited August 29, 2008 by MrGamma

[D H]

Okay.... It's tough for me to understand Latex at this point... in php...

Your php script is wrong. Force is mass times acceleration, not acceleration divided by mass.

When you come up with...

 

9.8\,m/s^2

 

according to the square it's the rate of acceleration...

 

9.8\,m/s^2 (first second)

19.6\,m/s^2 (second second)

39.2\,m/s^2 (after three seconds)

You have done something very wrong here. All it takes to compute the acceleration (*not the "rate of acceleration") is to combine Newton's law of gravitation and Newton's second law using the equivalence principle:

 

[math]

\begin{aligned}

F &=\frac{Gm_1m_2}{r^2} \\

a_1 &= \frac{F}{m_1} = \frac{Gm_2}{r^2} \\

a_2 &= \frac{F}{m_2} = \frac{Gm_1}{r^2}

\end{aligned}

[/math]

 

My main issue seems to be rooted in the classic equivalence principle. Where it says that mass does not effect acceleration. Objects falling to earth will do so at the same velocity. I imagine this is typically done because the effects are insignificant. However... for the sake of understanding this properly... This mass effecting acceleration is still hypothetical correct?

 

The article you cited is written for layman, and is a bit oversimplified. Better stated, the equivalence principle asserts that inertial mass is equal to gravitational mass. Gravitational mass: In the equation [math]F =Gm_1m_2/r^2[/math], the quantities [math]m_1[/math] and [math]m_2[/math] are the gravitational masses of the two objects. Inertial mass: in the equation [math]F=ma[/math], the quantity [math]m[/math] is the inertial mass of the object undergoing acceleration. What I did above in computing the acceleration due to gravity given the force due to gravity works if and only if gravitational mass and inertial mass are always the same.

 

It is my goal to understand orbital mechanics and I am currently under the belief that they do not use the mass of the object to determine acceleration. Which has me wondering... how do they know how to navigate their satellites?

 

What to they do? Just throw on the accelerator and wait for a ping back? If they gave a little bit too much throttle they play it by ear and adjust?

What they do is actually trickier than that. Accelerometers measure the net acceleration that results from the external forces acting on an object except for gravity. There is no way to directly measure the acceleration due to gravity! How space vehicles get from one place to another, in a nutshell:

• Analysts develop a trajectory plan that gives a broad plan of how the vehicle is to move over time.
  
• The spacecraft's navigation system estimates the vehicle's state (position, velocity, attitude, attitude rate) by combining readings from navigation sensors and models of the gravity fields. The inertial navigation system (INS) must use a mathematical model of the gravitational field because, as I mentioned above, it is impossible to measure the acceleration due to gravity. Because the state is numerically integrated using sensed accelerations plus estimated gravitational accelerations, the modeled state will drift from the true state over time. To counter this drift, the vehicle needs some measurements of the true state to update the estimated state.
  
• The spacecraft's guidance system compares where the vehicle should be based on the trajectory plan versus where it is based on the navigation system.
  
• The spacecraft's control system makes the vehicle get back on track.

[MrGamma]

Your php script is wrong. Force is mass times acceleration, not acceleration divided by mass.

 

Whoops... ???

 

function acceleration($m,$f,$r){
   return $f / ( ($m*2) / ($r*2) );
}

$f = 735; // Newtons...
$m = 75; // Persons Number One Weight in KiloGrams...
$r = 1; // Radius in Meters...
$acceleration = acceleration($m,$f,$r);

echo $acceleration;  

 

What they do is actually trickier than that. Accelerometers measure the net acceleration that results from the external forces acting on an object except for gravity. There is no way to directly measure the acceleration due to gravity!

 

What other forces besides the crafts own propulsion would effect it's acceleration?

 

How do I figure out how many newtons there are? Is 735 Newtons the gravitational constant of the earth?

[D H]

Whoops... ???

Still wrong. Acceleration is not force/mass²/distance². Look at the units! Force has units of MLT^-2, so your equation has units of M^-1L^-1T^-2. Use newton's second law: you know the mass and the force; acceleration is just force/mass.

 

What other forces besides the crafts own propulsion would effect it's acceleration?

Solar radiation pressure, aerodynamic drag, to name a couple.

 

How do I figure out how many newtons there are? Is 735 Newtons the gravitational constant of the earth?

735 newtons is the gravitation force between a 75 kg person on the surface of the Earth and the Earth itself.

[swansont]

mod note: moved from speculations to general physics

 

This is all pretty standard stuff under discussion.