1. Algebraic Statements And Scaling

2. Newton’s Laws of Motion (Axioms) 1.Every body continues in a state of rest or in a state of uniform motion in a straight line unless it is compelled to change that state by forces acting on it (law of inertia) 2.The change of motion is proportional to the motive force impressed (i.e. if the mass is constant, F = ma) 3.For every action, there is an equal and opposite reaction (That’s where forces come from!)

3. Newton’s Laws a) No force: particle at rest b) Force: particle starts moving c) Two forces: particle changes movement Gravity pulls baseball back to earth by continuously changing its velocity (and thereby its position)  Always the same constant pull

4. Law of Universal Gravitation Force = G M earth M man / R 2 M Earth M man R

5. Orbital Motion

6. Cannon “Thought Experiment” http://www.phys.virginia.edu/classes/109N/more_stuff/Appl ets/newt/newtmtn.htmlhttp://www.phys.virginia.edu/classes/109N/more_stuff/Appl ets/newt/newtmtn.html

7. From Newton to Einstein If we use Newton II and the law of universal gravity, we can calculate how a celestial object moves, i.e. figure out its acceleration, which leads to its velocity, which leads to its position as a function of time: ma= F = GMm/r 2 so its acceleration a= GM/r 2 is independent of its mass! This prompted Einstein to formulate his gravitational theory as pure geometry.

8. Applications From the distance r between two bodies and the gravitational acceleration a of one of the bodies, we can compute the mass M of the other F = ma = G Mm/r 2 (m cancels out) –From the weight of objects (i.e., the force of gravity) near the surface of the Earth, and known radius of Earth R E = 6.4  10 3 km, we find M E = 6  10 24 kg –Your weight on another planet is F = m  GM/r 2 E.g., on the Moon your weight would be 1/6 of what it is on Earth

9. Applications (cont’d) The mass of the Sun can be deduced from the orbital velocity of the planets: M S = r Orbit v Orbit 2 /G = 2  10 30 kg –actually, Sun and planets orbit their common center of mass Orbital mechanics. A body in an elliptical orbit cannot escape the mass it's orbiting unless something increases its velocity to a certain value called the escape velocity –Escape velocity from Earth's surface is about 25,000 mph (7 mi/sec)

10. Scaling Often one is interested in how quantities change when an object or a system is enlarged or shortened Different quantities will change by different factors! Typical example: how does the circumference, surface, volume of a sphere change when its radius changes?

11. How does it scale? Properties of objects scale like the perimeter, the area or the volume –Mass scales like the volume (“more of the same stuff”) –A roof will collect rain water proportional to its surface area

12. Example from Homework: Newton’s Law of Gravity Note that in order to compute a "factor of change" you can ask: by what factor do I have to multiply the original quantity in order to get the desired quantity? Example: Q: By what factor does the circumference of a circle change, if its diameter is halved? A: It changes by a factor 1/2 = 0.5, i.e. (new circumference) = 0.5 * (original circumference), regardless of the value of the original circumference. If the mass of the Sun was bigger by a factor 2.7, by what factor would the force of gravity change?  scales linear with mass  same factor If the mass of the Earth was bigger by a factor 2.2, by what factor would the force of gravity change?  scales linear with mass  same factor If the distance between the Earth and the Sun was bigger by a factor 1.2, by what factor would the force of gravity change?  falls off like the area  factor 1/ f 2 = 1/1.44 = 0.694