Gravity is the name associated with the mishaps of the milk spilled from the breakfast table to the kitchen floor and the youngster who topples to the pavement as the grand finale of the first bicycle ride.

Gravity is the name associated with the reason for "what goes up, must come down."

We all know of the word gravity - it is the thing that causes objects to fall to Earth.

The role of physics is to do more than to associate words with phenomenon. We need to explain phenomenon in terms of underlying principles. To explain phenomenon in terms of principles that are so universal that they are capable of explaining more than what we know gravity to be and more how we can describe the presence of gravity on a more universal level.

We will need to understand gravity along more complex lines and demonstrating this understanding by answering questions such as: its cause, its source, and any far-reaching implications on the structure and the motion of the objects in the universe.

Gravity is a force that exists between the Earth and the objects that are near it. As you stand upon the Earth, you experience this force. We have become accustomed to calling it the force of gravity. **How do we represent this as a symbol?**

This same force of gravity acts upon our bodies as we jump upwards from the Earth. As we rise upwards after our jump, **what does the force of gravity do to our upward speed?**

As we fall back to Earth after reaching the peak of our motion, **the force of gravity then does what in terms of speed?**

The force gravity causes an acceleration of our bodies during this brief trip away from the earth's surface and back. The acceleration of gravity is not to be confused with the force of gravity (Fgrav). The acceleration of gravity (g) is the acceleration experienced by an object when the only force acting upon it is the force of gravity. **What is this also know as? What is this in the absence of?**

On and near Earth's surface, the value for the acceleration of gravity is approximately **???????**. It is the same acceleration value for all objects, regardless of their mass (and assuming that the only significant force is gravity).

The following questions will be your task for completion for tomorrow's lesson in my absence. These will appear in google classroom and will be a daily grade

By whom was gravity discovered and what is the legend surrounding this discovery?

How can we describe the force of gravity?

How can proximity influence gravitational pull?

What variables affect the actual value of the force of gravity?

Provide an example of the presence of gravity in the world around us.

Why is the force of gravity on earth much stronger than the force of gravity on the moon?

How does gravity affect objects that are far beyond the surface of the Earth?

How far-reaching is gravity's influence?

Is the force of gravity that attracts my body to the Earth related to the force of gravity between the planets and the Sun?

These are the questions that will be pursued. If you can successfully answer them, then the sophistication of your understanding has extended beyond the point of merely associating the name "gravity" with falling phenomenon.

**Gravity Continued**

In the early 1600's, German mathematician and astronomer Johannes Kepler mathematically analyzed known astronomical data in order to develop three laws to describe the motion of planets about the sun. Kepler's three laws were developed off the data collected by his Danish predecessor and teacher, Tycho Brahe.

Kepler's three laws of planetary motion can be briefly described as follows:

The paths of the planets about the sun are elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses)

An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)

The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)

Kepler's third law - sometimes referred to as the law of harmonies - compares the orbital period and radius of orbit of a planet to those of other planets. Unlike Kepler's first and second laws that describe the motion characteristics of a single planet, the third law makes a comparison between the motion characteristics of different planets. The comparison being made is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every one of the planets. As an illustration, consider the orbital period and average distance from sun (orbital radius) for Earth and mars as given in the table below.

Kepler's laws provided a suitable framework for describing the motion and paths of planets about the sun, there was no accepted explanation for why such paths existed. The cause for how the planets moved as they did was never stated. Kepler could only suggest that there was some sort of interaction between the sun and the planets that provided the driving force for the planet's motion.

Newton was troubled by the lack of explanation for the planet's orbits and knew there must be some cause for such elliptical motion. Newton knew that there must be some sort of force that governed the heavens; for the motion of the moon in a circular path and of the planets in an elliptical path required that there be an inward component of force.

Circular and elliptical motion were clearly departures from the inertial paths (straight-line) of objects. And as such, these celestial motions required a cause in the form of an unbalanced force. The nature of such a force - its cause and its origin - bothered Newton for some time and was the fuel for much mental pondering. According to legend, a breakthrough came at age 24 in an apple orchard in England. Whether it is a myth or a reality, the fact is certain that it was Newton's ability to relate the cause for heavenly motion (the orbit of the moon about the earth) to the cause for Earthly motion (the falling of an apple to the Earth) that led him to his notion of universal gravitation.

**Newton's Mountain Thought Experiment**

As Newton started to study this force we call gravity, he viewed the moon as a projectile. Newton's reasoning proceeded as follows. Attempt to follow and understand Newton's standard of reasoning. Suppose a cannonball is fired horizontally from a very high mountain in a region devoid of air resistance. In the absence of gravity, what path would the canon ball maintain? In the presence of gravity, what path would the canon ball maintain?

Now suppose that the cannonball is fired horizontally again, yet with a greater speed. What would happen to the path of the canon ball in the presence of gravity? How would time affect the path of the canon ball?

Finally, suppose that there is a speed at which the cannonball could be fired so that the trajectory of the falling cannonball matched the curvature of the earth? If such a speed could be obtained, then the cannonball would fall around the earth instead of into it. The cannonball would fall towards the Earth without ever colliding into it and subsequently become a satellite orbiting in circular motion. Newton also reasoned that at even greater launch speeds, a cannonball would once more orbit the earth, but in an elliptical path.

The motion of the cannonball orbiting to the earth under the influence of gravity is an analogue to the motion of the moon orbiting the Earth. If the orbiting moon can be compared to the falling cannonball, it can even be compared to a falling apple. The same force that causes objects on Earth to fall to the earth also causes objects in the heavens to move along their circular and elliptical paths.

Newton's next dilemma was to provide reasonable evidence for the extension of the force of gravity from earth to the heavens. The key to this extension demanded that he be able to show how the affect of gravity is diluted with distance. It was known at the time, that the force of gravity causes earthbound objects (such as falling apples) to accelerate towards the earth at a rate of 9.8 m/s2. It was also known that the moon accelerated towards the earth at a rate of 0.00272 m/s2. If the same force that causes the acceleration of the apple to the earth also causes the acceleration of the moon towards the earth, then there must be a plausible explanation for why the acceleration of the moon is so much smaller than the acceleration of the apple. What is it about the force of gravity that causes the more distant moon to accelerate at a rate of acceleration that is approximately 1/3600-th the acceleration of the apple?

What can you conclude about this dilemma?

**Complete your worksheet to finish the lesson**. Tomorrow we will study the mathematical reality, intrinsic to the force of gravity that causes it to be inversely dependent upon the distance between the objects?

**From yesterday's lesson**, you should have been able to solve Newton's quandary based off the variable of proximity and comparison of the distance from the apple to the center of the earth with the distance from the moon to the center of the earth. The moon in its orbit about the earth is approximately 60 times further from the earth's center than the apple is. The mathematical relationship becomes clear. The force of gravity between the earth and any object is inversely proportional to the square of the distance that separates that object from the earth's center. The moon, being 60 times further away than the apple, experiences a force of gravity that is 1/(60)2 times that of the apple. The force of gravity follows an inverse square law.

The relationship between the force of gravity (Fgrav) between the earth and any other object and the distance that separates their centers (d) can be expressed by the relationship as stated on the board.

Since the distance d is in the denominator of this relationship, it can be said that the force of gravity is inversely related to the distance. Since the distance is raised to the second power, it can be said that the force of gravity is inversely related to the square of the distance. This mathematical relationship is sometimes referred to as an** inverse square law **since one quantity depends inversely upon the square of the other quantity. The inverse square relation between the force of gravity and the distance of separation provided sufficient evidence for Newton's explanation of why gravity can be credited as the cause of both the falling apple's acceleration and the orbiting moon's acceleration.

The inverse square law proposed by Newton suggests that the force of gravity acting between any two objects is inversely proportional to the square of the separation distance between the object's centers. Altering the separation distance (d) results in an alteration in the force of gravity acting between the objects. Since the two quantities are inversely proportional, an increase in one quantity results in a increase or decrease (eliminate as appropriate) in the value of the other quantity. **That is, an increase in the separation distance causes a increase / decrease (eliminate as appropriate) in the force of gravity and a decrease in the separation distance causes an increase / decrease (eliminate as appropriate) in the force of gravity.**

In addition, the factor by which the force of gravity is changed is the square of the factor by which the separation distance is changed. So if the separation distance is** doubled **(increased by a factor of 2), then the force of gravity is **decreased** by a factor of **four** (2 raised to the second power). If the separation distance is **tripled** (increased by a factor of 3), then the force of gravity is **decreased** by a factor of **nine** (3 raised to the second power).

Time for some questions...

**Wednesday**

Newton was able to draw an important conclusion about the dependence of gravity upon distance. This comparison led him to conclude that the force of gravitational attraction between the Earth and other objects is inversely proportional to the distance separating the earth's center from the object's center.

Distance is not the only variable affecting the magnitude of a gravitational force.

Fnet = **m • a**

The force that caused the apple's acceleration (gravity) was dependent upon the mass of the apple. The force acting to cause the apple's downward acceleration also causes the earth's upward acceleration (normal force) (Newton's third law). The force of gravity acting between the earth and any other object is directly proportional to the mass of the earth which applies an upward force to any downward force, directly proportional to the mass of the object, and inversely proportional to the square of the distance that separates the centers of the earth and the object.

The UNIVERSAL Gravitation Equation

But Newton's law of universal gravitation enabled the determination that gravitation is universal. ALL objects attract each other with a force of gravitational attraction. Gravity is universal. This force of gravitational attraction is directly dependent upon the masses of both objects and inversely proportional to the square of the distance that separates their centers. Newton's conclusion about the magnitude of gravitational forces is summarized symbolically as shown on the board.

The value of G can be determined as follows. This will be looked at in more detail in another lesson.

G = 6.673 x 10-11 N m2/kg2

When the units on G are substituted into the equation above and multiplied by m1• m2 units and divided by d2 units, the result will be Newtons - unit of force.

Using the equation on the board, give the following sample questions a go....

Sample Problem #1

Determine the force of gravitational attraction between the earth (m = 5.98 x 1024 kg) and a 70-kg physics student if the student is standing at sea level, a distance of 6.38 x 106 m from earth's center.

The solution of the problem involves substituting known values of G (6.673 x 10-11 N m2/kg2), m1 (5.98 x 1024 kg), m2 (70 kg) and d (6.38 x 106 m) into the universal gravitation equation and solving for Fgrav. The solution is as follows:

Sample Problem #2

Determine the force of gravitational attraction between the earth (m = 5.98 x 1024 kg) and a 70-kg physics student if the student is in an airplane at 40000 feet above earth's surface. This would place the student a distance of 6.39 x 106 m from earth's center.

The solution of the problem involves substituting known values of G (6.673 x 10-11 N m2/kg2), m1 (5.98 x 1024 kg), m2 (70 kg) and d (6.39 x 106 m) into the universal gravitation equation and solving for Fgrav. The solution is as follows:

Now attempt the same sample questions, this time using the following equation:

#### Fgrav = mg

Credits:

Created with images by Pexels - "action adult checkered shirt" • asw909 - "Prague Dec 2004: Statue of Tycho Brahe and Johannes Kepler"