Gravitation Help: Newton's Laws, Moon & Earth, and F_g=m*G/R_e^2

[Cyrus]

Hi, I've been reading on the laws of gravitation, let me just say, WOW, how can a man like Newton think of something that obscure. SHEESH! Anyways, I can see how each body gravitates with an equal and opposite force towards each other. Now, if this is the case, how come the moon and the Earth are at a distance appart. According to this they should have small accelerations towards each other, and with time, they should increase the speed at which the get closer. If you consider the billions of years of our solar system, I wonder why they are not together yet. Also, if you look at the equation $$F_g = m_e *m* G / R_e^2$$. It equates this to w=mg. But i thought that this equation is used for point mass bodies, which my physics book says is the case in symetrical spherical bodies. Is it ok to use this equation in finding gravity of an object because it is so tiny in comparison to the size of the earth, it is a point particle too? Finally, the denominator causes some confusion. As the distance r from the Earth gets bigger, the gravitation decreases. But if you are at the center of the earth, there should be zero gravitation, but this equation shows infinite gravitation, am i just using this equation outside of its limitations? Thanks for helping me.

Cheers,

Cyrus Abdollahi

 

[chroot]

Hi, I've been reading on the laws of gravitation, let me just say, WOW, how can a man like Newton think of something that obscure. SHEESH! Anyways, I can see how each body gravitates with an equal and opposite force towards each other. Now, if this is the case, how come the moon and the Earth are at a distance appart. According to this they should have small accelerations towards each other, and with time, they should increase the speed at which the get closer. If you consider the billions of years of our solar system, I wonder why they are not together yet.

The Earth and Moon are constantly accelerating towards each other. The concept you're missing is that of orbital velocity. The Moon is not just falling straight down toward the Earth; it's moving at a significant speed sideways too. The Moon moves sideways fast enough that for every foot it falls toward the Earth, it also moves sideways enough to gain a foot in altitude due to the curvature of the Earth. The net result is a nearly-circular orbit where the Moon is freely falling around the Earth, but not losing any altitude.

Also, if you look at the equation $$F_g = m_e *m* G / R_e^2$$. It equates this to w=mg. But i thought that this equation is used for point mass bodies, which my physics book says is the case in symetrical spherical bodies. Is it ok to use this equation in finding gravity of an object because it is so tiny in comparison to the size of the earth, it is a point particle too?

From a distance, a gravitating body appears to have all its mass concentrated at its center of gravity.

Finally, the denominator causes some confusion. As the distance r from the Earth gets bigger, the gravitation decreases. But if you are at the center of the earth, there should be zero gravitation, but this equation shows infinite gravitation, am i just using this equation outside of its limitations? Thanks for helping me.

Only the mass interior to your radius matters. In other words, if you're halfway to the center of the Earth, only the inner half of the Earth's mass matters. The outer half actually all cancels out. When you get to the center, there is zero gravitational force, because there is zero mass interior to you.

- Warren

 

[zefram_c]

It can be shown through calculus that the net gravitational force exerted by a uniform spherical shell is:

a) the same as if all the matter were concentrated at its center, OUTSIDE the spherical shell
b) zero, INSIDE the spherical shell

So when inside the Earth, all the mass outside your radius no longer matters. At the exact center of the Earth, the net force is zero.

 

[BobG]

Hi, I've been reading on the laws of gravitation, let me just say, WOW, how can a man like Newton think of something that obscure. SHEESH! Anyways, I can see how each body gravitates with an equal and opposite force towards each other. Now, if this is the case, how come the moon and the Earth are at a distance appart. According to this they should have small accelerations towards each other, and with time, they should increase the speed at which the get closer. If you consider the billions of years of our solar system, I wonder why they are not together yet.

How ironic. At first glance at this problem, Newton had the same impression as you. Given an object circling another, larger object, the small object should slowly spiral into the larger object. This was Newton's original response to this problem - a response that got him totally lambasted by Robert Hooke (who just lived to flame other scientists and scholars). Newton was so stung by Hooke's response that he devoted years to proving an object would, indeed, slowly spiral into the larger. Obviously, he never proved that conjecture since it was totally wrong. Instead his efforts resulted in his laws of motion, the universal law of gravitation, and a whole new mathematical method (calculus).

The easy way to see why this is wrong is to create a fictional point which has all the Earth's mass concentrated within it. A stationary object is going to go from zero velocity to a very fast velocity as it accelerates towards the point. Since the point is fictional and the object can pass right through it, it will - it has built up momentum and velocity that can't just suddenly go to zero. Once it passes through the point, gravity will slow the object at the same rate that it sped it up. In other words, the object will eventually reach zero velocity at exactly the same distance away from the point that it started at, except on the opposite side of the point. Once at zero, the object again accelerates towards the point, building up velocity, passing through the point and slowing at the exact same rate, and so on and on, back and forth forever.

Orbits are the same, but the object has a tangential, or lateral, velocity thrown in as well. Changes the shape from a straight line to an ellipse, but the principal is the same - however fast the object is accelerated towards the Earth on one side of the ellipse (apogee to perigee), it is slowed down by the exact same rate on the other side of the ellipse (perigee to apogee). There's no net change.