Unit 5: Gravity and Rotational Motion. Brent Royuk Phys-109 Concordia University

Transcription

1 Unit 5: Gravity and Rotational Motion Brent Royuk Phys-109 Concordia University

2 Rotational Concepts There s a whole branch of mechanics devoted to rotational motion, with angular equivalents for distance, speed, acceleration, mass, force, momentum and energy. Angular Speed, ω is measured in degrees (or radians) per second, not meters per second. Rotational Inertia, I depends on shape as well as mass, and measures how hard it is to rotate something. The rotational equivalent for force is Torque, τ which is a twisting force that depends on how hard you apply a force and how far away from the rotational axis you push. Torque = force x lever arm 2

3 Rotational Concepts Angular Momentum, L is a conserved quantity. L b = L a I b ω b = I a ω a Pulling string while bob rotates Spinning skaters Rotating platform 3

4 Rotational Vectors Gyroscopic stability Football spirals Bullets Precession Gyroscope Diving airplanes Helicopters Motorcycle turns 5

5 Traveling in Circles A mass on a string Which way does the mass accelerate? Which way does the force act on the mass? Two Latin Words Centripetal Centrifugal 7

6 Newton and Gravity Another amazing scientific achievement. 1665, 23 years old...in November [I] had the direct method of fluxions...in May following I had entrance into the inverse method of fluxions. And the same year I began to think of gravity extending to the orb of the Moon...& thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly. All this was in the two plague years of For in those days I was in the prime of my age of invention & minded Mathematicks & Philosophy more than at any time since. Newton in an unsent letter to Pierre Des Maizeaux "The Principia is pre-eminent above any other production of human genius." Pierre-Simon Laplace "Taking mathematics from the beginning of the world to the time when Newton lived, what he has done is much the better part." Gottfried Leibniz "All that has been accomplished in mathematics since his day has been a deductive, formal, and mathematical development of mechanics on the basis of Newton's laws." Ernst Mach 9

7 The Law Newton s Law of Universal Gravitation Whack! Isaac Newton, you ve discovered gravity! What s universal about Newton s Universal Law of Gravitation? The apple and the moon The moon is made of stuff, just like the apple. 10

8 Gravity The Law: F = G m 1 m 2 d 2 G = 6.67 x N m 2 /kg 2 Why don t we ever notice attractions between two terrestrial bodies? Physical attraction on a love seat. Let each person be a uniform-density sphere Big G, Little g Mt. Everest g Everest = 9.78 m/s 2 Gravity is an Inverse Square Law What if you halve the distance between objects? Double it? 11

9 Gravity g shuttle = 9.08 m/s 2, 7.44% less than Earth s surface. 12

10 Cavendish s Torsion Balance Weighing the Earth Huh? 13

11 Cavendish s Torsion Balance John Walker s modern reproduction Bending Spacetime in the Basement 14

12 Gravity and Orbit Why are astronauts weightless in orbit? 15

13 Projectile Motion What path does the ball follow when dropped? 17

14 Projectile Motion Horizontal Launch What happens if you kick a ball off a cliff? 18

15 Projectile Motion Falling Comparison 19

16 Projectile Motion Horizontal Launch 20

17 Projectile Motion 23

18 Projectile Motion Maximum Range R = v2 sin 2θ g Estimate the range of a well-thrown baseball. 24

19 Projectile Motion R = v2 sin 2θ g 25

20 Projectile Motion 26

21 Projectile Motion Air Resistance Data: 100 mph at 60 o ; vacuum = 581 ft., air = 323 ft. How about the moon? 27

22 Orbits Rocket Science At earth s surface, orbital velocity would be 7.9 km/s. The size of the satellite doesn t matter. 500 km above surface, 7.6 km/s. Geosynchronous Orbit Altitude = 36,000 km, orbital period = 1 day. History What makes the planets go? Aristotle: Newton: The next question was - what makes the planets go around the sun? At the time of Kepler some people answered this problem by saying that there were angels behind them beating their wings and pushing the planets around in an orbit. As you will see, the answer is not very far from the truth. The only difference is that the angels sit in a different direction and their wings push inward. - Richard Feynman, Character of Physical Law 30

23 Geosynchronous Orbit 22,236 miles high Earth s radius: 3959 miles 32

24 Escape Velocity What is the escape velocity of the Earth? v escape = 2gR = 11 km/s 33

25 Weightless in Orbit? 34

26 Newtonian Limitations 39