Newton's Law of Universal Gravitation
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[edit] Introduction
Sir Isaac Newtown is a man famously known for his contribution to the advancement of classical mechanics in physics. His ideas on moving bodies and forces were expressed in his laws of motion and law of universal gravitation.
[edit] Laws of Motion
There are three specific laws in Newton’s Laws of Motion
[edit] First Law: also known as Law of Inertia
Every object continues in its state of rest or of uniform motion in a straight line, unless it is compelled to change the state that it was in by forces impressed upon it.
The law basically states two points. When the object is sill, not moving, it will remain that way until a force acts upon it. The other point is similar, if the object is in motion, the object will remain in motion, and not change in acceleration until a force is acted upon the object.
[edit] Second Law:
The acceleration produced by a particular force acting on a body is directly proportional to the magnitude of the force and inversely proportional to the mass of the body.
The force can be represented by this formula:
- Fnet= m * a
OR
F=force m=mass a=acceleration sigma=sum total of all the forces (net force) The tiny dash above force and acceration shows that it is a vector.
In the first case, Newton's First law, there is no force acting on the object, therefore the acceleration of the object is constant. When the object undergoes constant acceleration (0m/s²) it is in equalibrium. There is no net force or unbablanced force. In Newton's Second law, he gives the reason for what would happen if the object was not at equilibrium. If there is an unbalanced force on the object, the object would accelerate, its speed and/or its direction would change.
According to the law, the two factors that affect the acceration is the net force and the mass. If the net force is increased and mass remains the same, the acceleration would increase; if the mass of the object increases the acceleration of the object would decrease.
[edit] Third Law:
To every action there is always an opposed equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
The action-reaction can be presented by this formula:
- FA on B=-FB on A
Everything around us follow this law. For example, a pesron sitting down on a chair. To make it easier to understand, the earth's pull(gravity) will be neglected. The person's weight is the downward force and the force oppsite to that, holding up the person is the chair providing an equal and opposite force. In this case the chair is strong enough therefore able to hold up the person, but if the up-ward force of the chair was not strong enough, then the chair would break and the person would fall.
[edit] Law of Universal Gravitation
Many people know the story of Issac Newton sitting under an apple tree; when an apple fell on his head, he suddenly thought of the concept of gravity. It is actually much more complex than that; the situation brought about Newton's law of universal gravitation.
As much credit as we give Newton for discovering the law, Johannes Kepler should be praised for his work also. Kepler was the assistant of an astonomer, Tycho Brahe, which he spent many years analyzing the work of. Kepler was determinded to discover the relationship between the number, distance, and motion of the planets around the sun using geometery and mathematics.
[edit] Kepler's Laws of Planetary Motion
[edit] Kepler's First Law: on Orbits
Every planet moves in an elliptical orbit, with the Sun at one focus.
[edit] Kepler's Second Law: on Areas
As a planet moves in its orbit a line drawn from the Sun to the Planet sweeps out equal areas in equal time intervals.See diagram below.
The line joining the two bodies (the sun and the revolving object) sweeps equal areas in equal times.
The graphic precisely displays Kepler's second law. The area of every triangle formed is exactly the same area. Also notice as the planet or satellite moves closer to the sun, the speed of its orbit increases, but when it is located farther away from the sun, the speed in which the planet or satellite revolves around the sun is slower.
[edit] Kepler's Third Law: on Periods
It stated that if P is the period and M is the length of semimajor axis of a planet's orbit, then the ratio P²/M³ is the same for all the planets.
Table of values that verifies Kepler's third law.
The linearity of the line between the period and lenth of the semimajor axis verifies that relationship.
[edit] Newton's Law of Gravitation
Issac Newton witnessed the apple falling onto Earth, there is no way he could not have also notice the moon revolving around Earth. Why does these things occur? The Earth must have a force that pulls the moon towards its center; if it did not then moon would either be travelling at uniform straight line motion or not moving at all, following Newton's first law. Newton obsevations were that all objects near the Earth's surface fall towards the Earth's center because of a gravitational force. That Gravitational force was Universal!
In Newton's Universal Law of Gravitation, he states
- The gravitational force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them.
In other words, it can be summarized by the formula:
F=Force, m1=mass of first object, m2=mass of second object, r=distance between the two objects, and G=universal gravitational constant
a better visual:
http://resources.yesican-science.ca/orbits1/images/trans_gravity_eq1.png
The universal gravitational Constant(G) has the value,
- G = 6.67 x 10-11 N*m²/kg²
The force due to gravity is dependent on the mass of the objects and the distance between them. If the mass of the two objects are very tiny, the force is also very small. If the distance between the objects are very large, the force would also be very tiny. This would explain the reason for why the apple falls toward the Earth's surface much quicker than the moon; the distance between the earth and moon is much further than the distance between the apple and the Earth.
[edit] Newton's Gravitation Law Verifies Kepler's Laws
Newton eventually verified that Kepler's first two laws imply a law of Gravitation. Newton showed that objects travel in elliptical orbits because of Inverse Square Law. Since the Earth travels around the sun in an ellipes, during certain areas when the Earth is closer to the sun, the Earth would also travel faster.
Kepler's third law also follows Newton's Law of Gravitation. Since Kepler's second law states that the planet's orbit speed must be constant and travels in a circular motion, then the planet is in uniform circular motion. This can then be expressed as:
mv²/R =G*Mm/R² or v² = GM/R
G=Gravitational Force, M=Mass of Sun, R=Radius
The period of a planet's orbit is one revolution around the sun and that can be expressed as
T = 2πR/v
Then if we square both sides of the equation so that we can plug in v² = GM/R into the equation:
T² = 4π²R²/v² = 4π²R²/(GM/R) = (4π²/GM)R³
We can then rewrite it as:
T²/R³ = 4π²/GM
which is equivalent to P²/M³ proving Kepler's third law.
[edit] Sample Practive Questions
[edit] Newton's Law of Motions
[edit] Newton's Law of Universal Gravitation
[edit] Reference
Newton's First Law
Newton's Second Law
Newton's Third Law
Kepler's Laws
Newton's Law of Universal Gravitations1
Newton's Law of Universal Gravitation2
[edit] Resources
Gonick & Huffman The Cartoon Guide to Physics New York, NY: HarperCollins 1990
Zitzewitz, Paul W. Physics Principles and Problems Columbus, OH: Glencoe/McGraw-Hill 1999
www.glenbrook.k12.il.us
www.sparknotes.com
www.phy6.org/readfirst.htm
