We’ve argued the force of gravity is also directly proportional to the masses involved: G is a universal constant measured to be 6.67  N·m 2 /kg N·m 2 /kg 2

Henry Cavendish (1731 – 1810)

How irresistible is the gravitational force of attraction between a pair of us when 1 meter (center-to-center) apart? G (80 kg) (70 kg) (1 meter) 2 = G 5600 kg 2 m 2 = N F grav R

2R R R mm FF R R/2 Two objects of mass, m, separated by a center-to-center distance R are mutually attracted to one another by a force F. How strong is the attractive force between the other pairs of objects shown? A. ¼ FC. F E. 4F B. ½ FD. 2F F. other mm mm m2m2m 2m2m2m2m

Weight How hard does earth’s gravity pull on me? distance to the center of the earth If unbalanced by any normal force of support how fast would I be accelerated? F = ma, so:

How fast does the earth accelerate me? How fast does the earth accelerate YOU? How fast does the earth accelerate an apple?

The earth accelerates ALL objects near its surface at exactly the same rate: a = (6.67  N·m 2 /kg 2 ) 5.98  kg (6.38  10 6 m) 2 = m/sec 2 Actually we’ve mixed up the historical order of events somewhat. Newton’s ideas do in fact explain why all objects fall to earth at the same rate independent of their mass. But notice calculating that the acceleration rate is exactly 9.8 m/sec 2 required knowing the mass of the earth! How is THAT known?

R earth = 6.38  10 6 meters Navigators and surveyors had established a size for the earth: = 1.09  m 3 Density of some STUFF in the earth: Granite2.7  10 3 kg/m 3 Iron7.8  10 3 kg/m 3 Water1.0  10 3 kg/m 3 Which would suggest the mass of the earth lies somewhere between 1 – 8  kg It was of course Cavendish who could take:

and for any moon orbiting any planet: For any planet (or star or asteroid) the acceleration down toward its center is which means:

Which is how we’ve known the mass of Mars, Jupiter and Saturn without even going there! How do we know the mass of Venus? Mercury? How do we know the mass of the sun?

Johannes Kepler ( ) All planets move in ellipses, with the sun at one focus.

The undefined concepts in GEOMETRY collection of all points in a plane equidistant from a selected point called the center.

collection of every point in a plane whose distances from two fixed focii sums to the same constant total. 5 inches 3.5 inches To build an ellipse, pick any two points and a number bigger than their distance apart. 3.5 inches 7 > 5 1 inch

collection of every point in a plane whose distances from two fixed focii sums to the same constant total. To build an ellipse, pick any two points and a number bigger than their distance apart. 7 > = 7 3 inch 4 inch = 7

collection of every point in a plane whose distances from two fixed focii sums to the same constant total. To build an ellipse, pick any two points and a number bigger than their distance apart. 7 > 5 2 inch 5 inch = 7

speed just right speed too high speed too low The second (far) focus is empty! The second (near) focus is empty!

Oops! Speed way too slow!

A popgun fires a standard 40mm (2.7 gram) ping pong ball at the back of a stationary wagon. mv  mv The ball ricochets back with almost the same speed. A. zero momentum D. momentum 2mv. B. momentum ½mv. E. momentum 3mv. C. momentum mv. F. momentum 4mv. The wagon must recoil with

mv Now compare the case where the tennis ball solidly rebounds

mv …to the case where the tennis ball smashes into but imbeds itself

mv ? ? Which collision sends the wagon rolling forward with the greater speed? A. B. C. Either collision gives the wagon the same speed.

mv ? ? 60 gram tennis ball initially at 10 m/sec 60 kg wagon initially at rest

A spray of rapid fire will provide a steady pressure that can levitate this block, holding it back in place.

v = 500 m/sec A ball rebounds from the ground straight up at 500 m/sec. How high can it climb? (How high had it been dropped from?) So: Or just:

v = 500 m/sec A 500 m/sec projectile traveling straight up, is in the air 102 seconds and reaches a km altitude. v = 500 m/sec A 500 m/sec projectile traveling horizontally can cross a 12 meter lecture hall in: dropping only Crosses the room in nearly a straight line! Rising h=1 meter: reduces its speed by ~4 cm/sec.

CO 2 molecules of a refrigerator-cold bubble in a can of soda pop 887 mph Room temperature nitrogen N 2 molecules 1160 mph An enormous number of these invisibly small particles exert a ~constant pressure outward, against all the surfaces of their container: The round shape characteristic of a balloon confirms that this pressure is exerted in all directions.

A. just as hard as Answers to Clicker Questions Just another example of Newton’s law: Interactions always involve mutually equal but opposite forces. Slide 2 presents an argument as to why. R/2 mm 2R mm R m2m2m R 2m2m2m2m Compare each to: