Gravity is so ubiquitous that we have difficulty thinking about it. But Newton did think about it and devised a theory of gravitation that stood almost unchallenged from 1684 to 1916, more than 250 years.
The students should learn that gravity produces a force on matter. The force is proportional to the mass. With more difficulty, the students should see that gravity is produced by a mass. Finally, the students will discover that gravity obeys an inverse square law.
OK, so you came to class as Galileo only two lessons ago. But wouldn't it be great to come to this class as Sir Isaac Newton?
The session may be difficult for students to do without guidance. If you feel they can make progress without lots of guidance, by all means, let them work this session using the scientific method as a guide. However, the instructor can also perform these demonstrations and then guide the students through the scientific method.
Begin by dropping the bean bag and tossing the apple up and catching it. Wonder aloud why it always comes down. Feel its weight (force). (Be careful with your language here. Weight is the force caused by gravity. Mass exists without reference to weight.) Toss apples to students and let them feel its weight. From the last session, remind the students that the downward force produces an acceleration of the mass and it falls to the floor.
Hold the bowling ball just over the mat. Drop it. Set the bowling ball on the floor. (Hopefully it will not move). Ask why it does not move in the horizontal direction but it does move in the vertical. To emphasize the point, hang a weight on a rope. Show that a sideways force would make the rope deviate from the vertical so the force is downward.
Push a beanbag slowly off the end of the long table. Repeat but this time give it a little shove so that it moves horizontally as it fall. Repeat several more times, each time shoving it faster in the horizontal direction. Ask the students to mentally compare the fall time for each repetition. Use the dowel to shove the bag down the table even faster until it is going as fast as you can safely make it go. Ask about the speed in the horizontal direction (constant) and the speed in the vertical direction (increasing downward). As them to imagine what would happen as the speed if the beanbag increases to bullet speed, to rocket speed. See if the students can also remember independence of motion in perpendicular directions.
Use the Galileo apparatus to remind students of the Galileo session. Then increase the slope of the trough dramatically. Help the students remember that the position increasing as t2 implies a constant acceleration. Ask if the ball still obeys the same speed proportionality. If you are brave, try dropping a beanbag from more than two meters and see if the students can measure the position as it falls. Instead of a clock you will have to count very rapidly and regularly -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5.
Drop a more and less massive object at the same time. Note that they hit the floor at about the same time.
Now start stacking the identical weights as weighing them. As the weight goes up, help the students to understand the difference between mass and weight. The masses appear identical and the weight goes up with mass.
Quickly move one of the large weights back and forth in the horizontal direction. Spread the large weights among the students and let them do the same thing (safely). Note that your hands feel a horizontal force associated with acceleration but when there is no horizontal acceleration, there is no horizontal force. Note that there is always a vertical force. Quickly moving the weight in the vertical direction seems to add or subtract an acceleration related force to the gravitational force.
Review everything and produces these hypotheses.
If you are in your Isaac Newton mode, have fun with this section.
Newton was inspired to think of gravitation, not by apples falling, but by the motion of planets and moons in our Solar system. He came to the conclusion that matter is attracted to matter and the attractive force is what we call the force of gravity. We have discovered that the force of gravity is proportional to the mass of the weights. The weights are attracted to the Earth. Newton conjectured that the second mass, the Earth, is also attracted to to the weights with an equal force but in the opposite direction. So the force of gravity is proportional to both the mass of the weights and the mass of the Earth. That is, the force is proportional to the product of the two attractive masses.
Newton used Kepler's laws of planetary motion (Equal areas swept in equal times, the period2 of a planetary orbit is proportional to the orbital semi-major axis3, planetary orbits are elliptical with the Sun at one focus of the ellipse) and his newly developed Calculus to show that gravity exhibits a force proportional to the inverse square of distance between objects. Such a force cannot be tested by simple Earth-based using the Earth itself because the Earth is too large.
We can see that force must decrease with distance because if it did not, we could feel the gravity of Jupiter and our own Moon and the Sun. The question is how fast does the force of gravity decrease with distance. We can make a "lines of force" argument. Let's imagine that a large spherical mass (radius r1) has a certain density of lines of gravitational force, d1, at its surface. (We think of the lines of force as radiating in straight lines from the center of the sphere.) Then we create an imaginary sphere with a much larger radius, r2. We will have the same number of lines of gravitational force but they will be spread over a larger area and have a density of d2. If we think of the force of gravity as proportional the density of the lines of force, we see the following:
4πr12 d1 = 4πr22 d2
The equation above says that the area times the number of gravitation lines of force per unit area (or the total number of lines of force) is a constant. We can rewrite this equation as
d2 = d1 r12 / r22
That is, the density of gravitational lines of force is inversely proportional to the square of the radius, an inverse square law. CAUTION: This argument is not in any way rigorous but it suggests that the force of gravity might obey an inverse square law.
We have just developed a conjecture that the force of gravity between two particles can be described by an expression like
This expression is not a vector expression (as required for any force) but a scalar expression. The scalar expression is adequate for one dimensional problems.
We can change the expression from a proportionality to an equality if we include a constant factor, G, call the "Gravitational Constant".
Newton showed that any spherical object that meets some simple requirements is equivalent to a particle. That is, we can treat a large object, like the Earth, as a particle of small size, located at the center of the Earth.