Gravity

From Geophysics 300

Gravity wiki

1 History
2 Theory
3 Applications
4 Authors

History

To start off the History of Gravity, we need to know about the people who first had ideas about it.

ANCIENT

In Greek mythology the Earth was a disk-shaped region embracing the lands of the Mediterranean region and surrounded by a circular stream. Greek philosopher Anaximander visualized the heavens as a celestial sphere that surrounded a flat Earth at its center. (Fundamentals of Geophysics by William Lowrie)

It wasn't until Pythagoras started to speculate that the Earth was a sphere that the scientist began to look for proof of that. Aristotle than moved on that idea further. The reason why we bring this up, about the shape of the Earth, is that it is essential to the coming about of scientist trying to explain why the Earth is a sphere and why it revolves around the Sun.

MODERN

In 1687, English mathematician Sir Isaac Newton published Principia, which hypothesizes the inverse-square law of universal gravitation. In his own words, “I deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centers about which they revolve; and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the Earth; and found them answer pretty nearly.” Gravitation

Newton's law of gravity states that the force between two objects in the universe is equal to the product of the masses of the two objects divided by the square of the distance between the two objects. Author: de Rooij, Mark Source: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES A-STATISTICS IN SOCIETY Volume: 171 Pages: 137-157 Part: Part 1 Published: 2008

The following equation is used for universal gravitation:

$F = G * [(m 1) * (m 2) / (r 2)]$

This equation can be used anywhere in the universe, as long as you know three of the unknown variables, the fourth can be calculated. G is a constant and therefore is already known.

For more information about the history of gravity, see:

History of Gravitational Theory

Physics by John D. Cutnell and Kenneth W. Johnson

Last, to get a more personal history of the people behind gravity read:

Einstein's Heroes: Imagining the World through the Language of Mathematics by Robyn Arianrhod.

Theory

A very long time ago was a man that we know as 'Newton.' He developed his Law of Universal Gravitation by finding that the attracting force between two masses (M and m) was inversely proportional to the distance (r) squared between them times the universal gravity constant (G, there is more on 'G' below):.

Now, it is possible to rewrite this equation to solve for the 'gravity' an object feels on any planet with a known mass (Mp) and a known radius (rp): .

Since, most of us live on Earth, let's use this equation to see what average gravity is on the surface of the Earth. Earth's average radius is 6.37*10^6 m and it weighs about 5.98*10^24 kg: (I have added an extra 'kg' to the top of the equation to make the units cancel).

The Earth is not a perfect sphere as it has mountains and valleys, and it is wider in the middle due to centrifugal force at the equator as it spins on its axis. Therefore, if you live on the poles you actually weigh more than you would if you lived at the equator. The gravity difference between the poles and the equator is:

Where the difference in gravity at the equator is:

and the velocity at the equator is .

Note: this is assuming the Earth is a perfect sphere.

There are ways to correct for 'anomalies' that we experince on this imperfect Earth, and here is one of them, free air correction. It is possible to correct for other gravity anomalies. This page was developed by other students at Boise State and also deals with the anomaly problem (midway down the page). (Click links for more information).

Time went by and other scientists discovered ways to verify the fact that gravity is about 9.81 m/s^2 on Earth and here are some of the devices that they used.

Unbalanced Forces:

Introduction

One day Atwood connected a smaller mass to a larger mass with a string. He strung the string over a pulley above both masses and was able to use this contraption to verify that gravity was really 9.8 m/s^2. He probably used some equations similar to the ones described in the theory section below.

Theory

This lab focused on the machine developed by Atwood, hence the name Atwood’s machine. A smaller mass (M1) is connected to a larger mass (M2) by an ‘ideal string’ that goes over an ‘ideal pulley.’ Since M2 is greater than M1, M1 will accelerate upward as M2 accelerates downward and since the two are connected, they will accelerate at the same rate. This acceleration (a) can be determined by where Δy is the distance M2 fell. Once ‘a’ is found, it can be used to determine gravity by .

Newton’s 2nd:

Introduction

I am not sure who was the first person to design and test this apparatus. It basically consists of a larger mass that slids along on its belly as a smaller object connected to it by a string looped over two pulleys pulls it forward. Kinetic friction resists motion and points in the direction opposing the direction of motion. Again, the equations that he used to verify gravity had to be similar to the ones we used in the theory section.

Theory

This lab was similar to the unbalanced forces lab except it had a larger mass M connected to a smaller hanging mass m by a string that went over two pulleys (one was the level of the table, and the other was elevated). ‘M’ slid along on a level surface and we had to determine the coefficient of kinetic friction (μk) by elevating one end of the ramp and using this equation: where θ is the angle of the ramp determined by: θ = arctan(height of ramp 1 meter away from base) and ‘a’ was determined using the equation from Unbalanced Forces as the object slid one meter down the ramp. This time ‘g’ was determined by: .

Angular Acceleration:

Introduction

Galileo once rolled masses down a smooth incline to discover that they, regardless of mass, rolled at the same rate. You may be familiar with the force equivalent of Newton's Second Law, F=ma. There is another form of this equation that involves torque. That is and is described in full detail below in the theory section.

Theory

This lab used the torque (τ) equivalent of Newton’s second law: where I is the moment of inertia and α was angular acceleration. A rotating object requires torque to change its angular acceleration. You can use an object’s mass (M) and geometric distribution to determine its moment of inertia (I). It is possible to get tangential acceleration (a) from angular acceleration by where r is the radius of rotation (or the radius from the outside to the center of the object). Objects roll around an axis through the point of contact and their moment of inertia depends on this. Their acceleration is also dependent on their moment of inertia. The “parallel-axis theorem” makes it possible to move the moment of inertia to the center-of-mass (Icm) and is (M is mass of the object as described above, and R is the radius of the object). Note that R is also the height from the center of mass to the point of contact so R is sometimes replaced by h.

There is a way to determine how fast an object accelerates down an incline without knowing its mass or its radius. All you do is time how long the object takes to roll down an incline (t) angled with a known θ.

The equation is where s is the distance an object rolled.

Something to think about

Ever wanted to know who determined 'G?' The Universal Gravity Constant 'G' was indirectly determined by Henry Cavendish who used Newton's Law of Universal Gravitation to develop his 'machine.'

Works Cited

Knight, Randall D. "PHYSICS FOR SCIENTISTS AND ENGINEERS SECOND EDITION." San Francisco, California: Pearson Education, 2008.

Rienmann, R. J. and Luke, R. A. "Cooperative Exercises in Phyiscs I" Boise, Idaho: Boise State University, 2007.

Applications

Earth-based applications

A first application of the Universal law of Gravity is the way each object on Earth is attracted towards the center of our planet. This as however been known for a long time, as each human being is subjected to it. The gravitational force is used in number of applications like hydroelectric plants or rollercoasters for instance.

Another application, perhaps more interesting is the fact that local gravity at the Earth's surface is influenced by density anomalies in the shallow subsurface. This allows geophysical exploration using the gravity in order to detect geological features underground, like basalt, buried lava flows...

Gravity surveys are performed by measuring the relative gravity field at various point in reference to the gravity at a base, which has a null relative gravity. The measurements are then corrected for the elevation ("free-air" effect), the topography: additional mass below the point of measurement (Bouguer correction), the latitude: the gravity decrease towards the poles because of the earth rotation. The resulting corrected measurement allow to detect density anomalies. These can be local, in the shallow subsurface, and are caused by small geologic features or ore bodies. On a larger scale, gravity data can help determining the crustal structure of the earth. The crust has indeed a lower density than the mantle, so gravity anomalies can be related to the crust thickness. Finally, on an even bigger scale, gravity can be used to measure tidal effects, as well as mantle currents and mantle rotation. This characterization of the earth is part of a branch call Geodesy of earth sciences.Geodesy on Wikipedia

Space applications

Geostationary orbit

The main consequence of the Universal law of Gravity is that the planetary motions are dictated by the attraction between spatial bodies. It becomes then possible to predict very precisely the trajectory of planets, (XVII century). It is however now widely used for artificial satellites and space exploration. Accurate computing of the trajectories is now possible by taking in account the mass and position of spatial bodies, and allowed to send probes as far as Neptune just by using the attraction of planets to "launch" the probes further, once they have escaped the Earth gravity. The gravitational force between planets is responsible for anomalies on there ideal trajectories as if they were only attracted to the Sun (two bodies problem). Anomalies on Neptune's trajectories thus allowed to predict the existence of Pluto before it was actually seen. Moreover, the influence of planets on their associated star's trajectory can now be detected and leaded to the discovery of extra solar-system planets, also called exoplanets.

The Law of Gravity brought to light a specific orbit around the spinning Earth, called geostationary orbit. This very unique orbit is defined by the fact that a satellite in this orbit is apparently staying at the same position in the sky when viewed from the Earth. This orbit is in the equatorial plane, and its altitude can be calculated with Kepler's third law of motion:

$\frac{r_s^3}{T_s^2}=\frac{GE}{4\pi^2}$

Where G the universal gravitational constant and E the mass of the Earth. We want to have $T s$ equals to the Earth's sidereal period (the amount of time needed for a 360 degree rotation), which is 86164 s. The resulting radius is approximately 42,165 km, corresponding to an altitude of 35,786 km above sea level at the equator. The main advantage of such a satellite is that it is constantly "looking" at the same part of the earth, making it very useful for broadcasting TV or carrying communications over a specific region for example. This orbit is also used for meteorological satellites.

As satellites trajectories are linked to the local gravity field, the precise monitoring of a satellite position (on a low orbit) can be used to obtain its local gravity field and from the to map the earth gravity, or another celestial object around which the satellite is orbiting.