Chapter 12: Universal Gravitation


Definitions:
12:2: tangential velocity: A component of velocity tanget to the trajectory of a projectile.
12:4: law of universal gravitation: For any pair of objects, each object attracts the other object with a force that is directly porportional to the product of the masses of the objects, and in inversely proportional to the square of the distance between their centers of mass.
12:4: universal gravitation constant: The constant, G, in the equation for Newton's law of universal gravitation.
12:5: inverse square law: A physical quantity varies inversely as another quantity is squared.
12:6: perturbation: the changing of a planet's tradition orbit due to the influence of gravitational forces.

All definitions taken from Conceptual Physics: Third Edition.



12:1 The Falling Apple
Issac Newton is credited with discovering the the idea of gravity is applicable throughout the universe. The common myth is that Newton was sitting under an apple tree, when an apple fell and hit him on the head. Newtown knew about the properties of interia, and also knew that if moving objects don't have an outside force acting on them, they continue moving in a straight line and at the same speed. Since the apple that hit him changed speed and direction, he knew there must have been a force acting on it. Through this realization, Newton saw that the moon is falling towards the Earth and he figured that the moon and the apple were both affected by Earth's gravity.
This picture, from http://encyclozine.com/History/Biography/Newton/Newton.jpg, depicts Issac Netwon, who discovered the affects of gravity in the universe.
This picture, from http://encyclozine.com/History/Biography/Newton/Newton.jpg, depicts Issac Netwon, who discovered the affects of gravity in the universe.

12:2 The Falling Moon
Newton realized that since the moon doesn't move in a straight line, then it must be falling around the Earth. He reasoned that it is falling around the Earth because it was moving below the straight line it would follow had there been no outside force. He guessed that the moon was a projectile around the Earth because of gravity. Newton compared the falling of the moon to that of a high-speed cannon fired above the Eahrth's atmosphere. He believed that the cannon, if fired high enough and fast enough, would react exactly as the moon and fall around the Earth. If the cannon was fired with a low horizontal speed, however, it would form a parabolic path and eventually hit the Earth. The faster the cannon would be fired, the less curved the path it traveled would be. Both the moon and the cannon have velocity parallel to the Earth's surface, called tangential velocity. This velocity guarentees an almost circular path around the Earth, instead of into it.
Newton had to find a way to test his hypothesis and test if the moon's fell beneath the straight-line path in proportion with any object at the Earth's surface, such as an apple. He thought mass wouldn't affect the way an object fell because mass doesn't affect accelartion of free-falling objects on Earth. The distance the moon and the apple fall should only correspond to the distance from the Earth's center and nothing else. If the distance is in correct proportion, then the idea that the Earth's gravity reaches the moon would be proven correct.
Newtown knew that the moon was 60 times farther from the center of the Earth than the apple at the center of the Earth's surface. The apple would fall 4.9 m in it's first second of free-fall. Because of this, Newton figured gravitation attraction would be influenced by relative distance. From this, Newton realized that in one second, the moon would fall 1.4 mm. This value representes how far the moon's orbit was below the straight line distance it would follow if not affected by gravity, which was about the same as today's accepted value. Because Newtown was unsure of the exact distance from the Earth to the moon, his findings were harshly criticismed, and he put his findings in a drawer and kept them there for 20 years. In order to prove his findings, Newton invented a new section of math, Calculus. He then published the Law of Universial Gravitation, saying that all objects in the universe attract one another.
Above is Newton's original drawing explaining how a projectile fired fast enough, like the moon, would fall around the Earth.  This image is taken from http://ircamera.as.arizona.edu/NatSci102/NatSci102/images/newtmtn.gif
Above is Newton's original drawing explaining how a projectile fired fast enough, like the moon, would fall around the Earth. This image is taken from http://ircamera.as.arizona.edu/NatSci102/NatSci102/images/newtmtn.gif


12:3 The Falling Earth
Newton's Law of Universal Gravitation confirmed that the Earth was not the center of the universe. His theory of gravitation explained the rotation of the planets around the sun, showiing that the planets fall around the sun just like the moon falls around the Earth. The planets also have tangential velocities that prevent them from crashing into the sun.

12:4 Newton's Law of Universal Gravitation
Newton only discovered that gravity is universal, not the conept of gravity. Newton's law of gravitation says that every object attracts eery other object with a force that for any two objects is directly proportional to the mass of each object. Newtown discovered that the force decreases as the square of the distance between the mass of the objects increase. The farther the distance between the objects, the lower the force between them.
Mathematically, the law is expressed as F ~ (m1m2)/d^2 where d is the distance between the centers of the two masses, m1 is the mass of the first object , m2 and is the mass of the second object.
The univerasl law of gravitation can be expressed as an exact euqatil with a constant as well. The constant, G, is the universal gravitation constant. Then, the equation is
F = G (m1m2)/d2 . The equation means that to find the force of gravity between two masses, then dividing by the square of the distances between them, and then multiplying that value by G. The value of G ensures that the force is given in Newtons when plugged into the equation. The value of G in scientific notation is equal to 6.67 X 10^11 (N(m^2))-kg^2. Because the value of G by itself is such a small value, it signals that the force of ravity is extremely weak. It is the weakest of the four fundamental forces, and is only felt when large masses are involved. The force of attraction between you and the Earth is your weight. Aside from mass, your weight also depends on the distance from the center of the Earth. The further from the center of the Earth you are, the less you weigh, but your mass always remains the same.

12:5 Gravity and Distance: The Inverse Square Law
The inverse square law shows that if an object moves twice as far, the force is one forth as much. Similarly, if it is three times as far, the force is one ninth, four times as far, one sixteenth, and so on. The Earth's gravity will never drop to zero no matter how far from the center you are. However, the Earth's gravitation may be overwhelmed by gravity from closer and bigger masses, but it will always still be there.

12:6 Universal Gravitation
The Earth is round because of gravitation. Any corners the Earth may have had previously have been pulled in because the Earth's gravity acted upon itself, making the it a sphere. The net force on Jupiter is not from the sun, but from all planets. However, their effects are less than those from the sun. This can be
seen when Saturn is near Jupiter and Saturn's pull on Jupiter disturbs its orbit, causing both planets to deviate from their normal path. This is called perturbation. In the middle of the 19th century, Uranus was the newest planet discovered in the solar system. Astronomers saw perturbations of Uranus' orbit, leading to the discovery of Neptune. This then led to the discovery of Pluto in 1930 for the same reason. The shapes of other galaxies prove that the law of gravitation extends beyond our solar system.
Above is a diagram of the solar system when Pluto was still considered a planet. As described, the orbit of Neptune was perturbed because of the gravitational pull from Pluto. This image is taken from http://www.ioncmaste.ca/homepage/resources/web_resources/CSA_Astro9/files/images/unit4/solar_system.jpg
Above is a diagram of the solar system when Pluto was still considered a planet. As described, the orbit of Neptune was perturbed because of the gravitational pull from Pluto. This image is taken from http://www.ioncmaste.ca/homepage/resources/web_resources/CSA_Astro9/files/images/unit4/solar_system.jpg