We build on Newton's Laws to develop the foundation for the rest of the course.
Some Applications of Newton's Laws
Key points: How Newton's Laws are applied; origin of tides; kinetic and potential energy; escape velocity
|An example of using the laws of motion and gravity:
Our experience is indicated on the left: an elephant and a feather fall at very different rates. The arrows show the accelerations, and in agreement they show that the feather has a much smaller acceleration than the elephant. However, experiments show that the two would have the same acceleration and fall at the same rates if there were no air resistance, as shown on the right. Here is a more dramatic demonstration of the same thing(reload to restart lecture animations). Can you use the laws you have just learned to explain this behavior??? (from The Physics Classroom, http://www.physicsclassroom.com/mmedia/newtlaws/cci.html and Central Valley Christian Schools, Visalika, CA, http://home.cvc.org/physics/apollompeg.htm)
If F = ma, then a = acceleration = F/m.
In , M is the earth mass and m the mass of the object attracted to the earth. If Forcegrav = F, we get acceleration = a = F/m = GM/r2 . The acceleration does not depend on the mass of the object. The force F=ma is the object's weight.
|Near or on the earths surface, the acceleration is essentially a constant because everything is at the same distance, r, from the center of Earth. We therefore frequently equate weight with mass although strictly speaking, weight is just a measure of the value of F in Newtons Second Law while mass is a property intrinsic to the object under discussion (from Journey Through Astronomy, http://csep10.phys.utk.edu/).|
|The lunar gravity on the near side of Earth is stronger than on the far side. As a result, the earth distorts. Water distorts much more than rock, so the water level rises on ocean shores. High tide occurs roughly twice per day, as the earth rotates to put the same face toward the moon and then opposite the moon (see diagram to left).(By G. Rieke)|
|The same effect occurs for the pull of the sun -- the largest tides are when moon and sun tug in the same direction. The tides appear roughly twice a day because the earth rotates under the fixed distortion imposed by Sun and Moon (Illustration by G. Rieke).|
|Here is how it looks in motion.
(By G. Rieke)
Newton's Laws give us precise definitions of energy, defined as the ability to do work, which in turn is defined as a force acting over a distance. Power is energy per second, so it reflects the ability to do continuous work. For example, the falling elephant above can definitely exert a force over a distance, and the associated energy before he starts to fall is called potential energy:
where M is the mass of the earth, m the mass of the elephant, and r his distance from the center of the earth. The potential energy is defined in a way that makes it nearly zero far from the earth, and increasingly negative as the elephant falls farther toward the center of the earth. Once he is falling, he also has kinetic energy
where v is his velocity.
|The law of conservation of energy requires that the sum of the elephant's potential and kinetic energy always stay the same, unless he actually does work to something else (like the ground when he hits it at the end of his fall). This animation shows total energy, E, divided between kinetic energy, KE, and potential energy, PE. When the ball is moving rapidly at the bottom of its travel, it has kinetic energy. When it is at the top of its motion, it has potential energy. However, the sum of the two is always the same. From Scott Anderson, copyright open course, http://www.opencourse.info/ (a fixed amount of potential energy may be assumed subracted off the animation, if the motion occurs from the surface of the earth - it makes no difference to the physical principle illustrated)|
Suppose we want to put the elephant into a cannon and shoot him into space? How much energy would we need to give him escape velocity (so he would escape the gravity of the earth and never come back)? If we use the least energy required, his motion will have come to zero just when he has escaped. Since his total energy is then zero, we can write:
where the potential energy is the least that puts him permanently into space. We can solve for the escape velocity:
Newtons Laws were tested in many ways, and are still used for most calculations of motion and of behavior involving gravity. However, around 1900, Einstein realized that there were some very subtle inconsistencies, and he developed his theories of relativity to deal with them. His postulates required a revision of Newtons Laws for objects moving at high speeds or in very strong gravitational fields.
For example, there was an unexplained discrepancy in the position of Mercury. Its perihelion, the point of closest approach to the sun was not a fixed position but was slowly moving, called precessing (from Journey Through Astronomy, http://csep10.phys.utk.edu/).
Because Mercury lies so close to the sun, it is moving through a region where the suns mass causes space to be curved enough to be noticed. Einstein used his theory to calculate what the precession of Mercurys orbit should be, and he matched the observed value, once again illustrating how the scientific method works. The idea of curved space provides a shorthand way to describe the effects of gravity on light as well as on objects with mass. Rather than traveling in straight lines, light follows lines that track the curvature of space.
Einstein also added a new concept to the Newtonian idea of energy: that mass and energy are equivalent in certain senses. Specifically, a mass, m, can be annihilated to yield an amount of energy
E = mc2
where c is the speed of light.
|The awesome (and frightening) power of a hydrogen bomb results from converting just a few ounces of matter into energy, according to E=mc2 (From The Hydrogen Bomb Homepage, http://www.bilderberg.org/hbomb.htm)|
Test your understanding before going on
Time magazine cover on scientific vs. religious approaches, from http://www.trinity.edu/mkearl/knowledg.html
A brilliant beam of light from an alien spaceship, from Steven Spielberg's "Close Encounters of the Third Kind," http://www.caiusfilms.com
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hypertext G. H. Rieke
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