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# Gravity

This review covers the principles of free fall, equations of motion, Newton's laws, and universal gravitation.

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## Gravity

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1. Gravity • Read Your Textbook: Foundations of Astronomy • Chapter 5 • Homework Problems • Review Questions: 3, 4, 5, 9, 10 • Review Problems: 1, 3, 4 • Web Inquiries:

2. Galileo • Galileo Discovered: • Objects fall at the same rate, independent of their mass. • The increasing rate of speed is uniform. • The distance they fall each second follows the odd numbers. • The total distance fallen is proportional to the time2.

3. Free Fall Measurement Time Distance Total Distance 1 1 1 Let 1 unit of distance = the distance the object falls during the first second. This turns out to be 4.9 m ~ 5 m The acceleration is uniform, g = 9.8 m/s/s ~ 10 m/s/s 1

4. Free Fall Measurement Time Distance Total Distance 1 1 1 2 3 4 3 5 9 1 3 5

5. Free Fall Measurement Time Distance Total Distance 1 1 1 2 3 4 3 5 9 4 7 16 1 3 5 7

6. Free Fall Measurement Time Distance Total Distance 1 1 1 2 3 4 3 5 9 4 7 16 5 9 25 1 3 5 7 9

7. Free Fall Measurement Time Distance Total Distance 1 1 1 2 3 4 3 5 9 4 7 16 5 9 25 … … … t2 1 3 5 7 9

8. Free-fall Velocity v = vo + a t

9. Equations of Motion v1= v0 +a t velocity x1= v0t + 1/2 a t2 distance Initial Conditions:Start at rest, v0 = 0 Acceleration is gravity a = g~ 10 m/s2

10. Free Fall Equations v1= v0 +a t x1= v0t + 1/2 a t2 Initial Conditions:Start at rest, v0 = 0 Acceleration is gravity a = g~ 10 m/s2 v1 = g t(This is how the velocity changes with time) x1 = 1/2 g t2(This is how total distance changes with time)

11. Reaction Time v1 = g t x1 = 1/2 g t2 SOLVE FOR TIME x1 = 1/2 g t2 2 x1 = g t2 2x1  g = t2 t = 2x1  g

12. Reaction Time Experiment t = 2x1  g a) Measure the distance that the ruler falls x1 in centimeters. b) Multiply by 2 c) Divide by 980 d) Square Root t = 2 x1 980 cm/s2

13. Acceleration of Gravity Things that fall, accelerate at 9.8 m/sec/sec near the Earth's surface. This means velocity of a falling body increases by 9.8 m/sec with each passing second. Acceleration is the change in velocity over the change in time. a = Dv/Dt

14. Horizontal and Vertical Motion

15. Projectiles • Galileo’s Trajectories x = vox t y = voy t - 1/2 g t2 The horizontal distance (x) is just due to the initial velocity in the horizontal direction (vox). Or, how much kinetic energy is imparted to the object.

16. Projectiles x = vox t y = voyt - 1/2 g t2

17. Trajectory Modified By Gravity Path in the absence of g x = vox t y = voyt - 1/2 g t2 1/2 g t2 1/2 g t2

18. Velocities x = vox t y = voyt - 1/2 g t2

19. Isaac Newton Father of Modern Physics Newton -- Born Dec 25, 1642 or Jan 4, 1643 (Pope Gregory's 11 days) Published Principia in 1687. Invented modern physics and calculus. This book had an immense effect on physics and astronomy.

20. Newton’s 1st Law Inertia An object at rest remains at rest OR if in motion, moves at constant velocity in a straight line UNLESS acted on by a net external force. Inertia : the property of matter that resists motion. Mass : the measure of inertia. Mass is an innate characteristic of any chunk of matter. Weight is actually the Force experienced by a Mass due to g.

21. Newton’s 2nd Law F = m a An acceleration, a, is produced when a force of magnitude F acts upon an object of mass m. Twice the force will give twice the acceleration. If you try to move twice the mass with a given amount of force you'll only produce half the original acceleration. Weight is the force felt by a mass due to acceleration. W = m g

22. Weight and Force Our weight (W) is an example of the force (F) we feel due to the acceleration of gravity (g). W = mg (F = ma)

23. Mass No Matter Lead and wood balls accelerate at the same rate when dropped from Pisa’s leaning tower. A hammer and feather fall at same rate in a vacuum. Apollo 15 astronauts tested Galileo's hypothesis on the Moon Astronaut David R. Scott, Apollo 15 commander, watches a geological hammer and a feather hit the lunar surface simultaneously in a test of Galileo's law of motion concerning falling bodies.

24. Newton’s 3rd Law For every action, there is an equal and opposite reaction. Objects do not just act, they interact. I pull on the Earth, it pulls on me.

25. Newton's Law of Universal Gravitation Fgravity = m GM/R2 This means that the force of gravity between any two bodies in the universe is equal to a constant (the Gravitational Constant, G = 6.67x10-11 N-m2/kg2) times the product of the masses of the two bodies in question (m and M), divided by the square of the distance between their centers (R).

26. Newton's Law of Universal Gravitation Fgravity = m GM/R2 Double the mass, double the force. Double the distance, reduce the force by 1/4. Triple both mass and distance?

27. Newton's Law of Universal Gravitation Fgravity = m GM/R2 Double the mass, double the force. Double the distance, reduce the force by 1/4. Triple both mass and distance? Reduce the force by 1/3. 3X from M, (1/3)2 from R

28. What Goes Up, Must Come Down Equating Newton's second law with gravity F = m a F = m GM/R2 m a = m GM/R2 m = apple, m = human, m = projectile, m = moon?

29. What Goes Up, Must Come Down Equating Newton's second law with gravity F = m a F = m GM/R2 m a = m GM/R2 a = GM/R2 Acceleration is GM/R2 , irregardless of the mass m.

30. Surface Gravity M F = ma F = mGM/R2 a = GM/R2 • All free-falling bodies accelerate uniformly independent of their mass. • Acceleration depends only on the mass of the attracting body and the distance from its center. • Earth’s Surface Gravity a = g = G Mearth/Rearth2 • g = 9.8 m/s/s R m

31. Gee, its “g” • g = 9.8 m/s2 • Surface Gravity BUT, note that it is dependent on r. Near the surface r = Rearth Want to lose weight? Hike to the top of a hill. Acceleration due to gravity will be less, therefore your weight will be less.

32. Lunar Surface Gravity • Mmoon = 0.123 Mearth ~ 1/10 • Rmoon = 0.270 Rearth ~ 1/4 • gmoon = G (0.123 Mearth)/(0.270Rearth)2 ~ 1/6 gearth

33. Moon Gravity • Moon’s Surface Gravity a = gmoon = G Mmoon/Rmoon2 • gmoon = 1.6 m/s/s • Weight on the moon, W = mgmoon • Since gmoon/gearth = 1/6, Wmoon/Wearth = 1/6 • You will weigh 1/6 as much, but your mass on the moon is the same as mass on the earth!

34. Measurement of Mearth "g" is called our surface gravitational acceleration, and g = 9.8 m/sec/sec . The value g depends on G (a constant), the mass of the earth (M) and the radius of the earth (R).

35. Cavendish g = GM/R2 M = gR2/G F = m GM/R2 GM = FR2/m Mearth = 5.976 x 1024 kg

36. Gravity Works Everywhere

37. Earth and Moon g = GM/R2 g = GM/(60R)2 acceleration of the moon due to earth g is 1/3600th as great at the moon than it is at the earth surface. If the distance fallen in 1 second is 4.9 meters at the surface of the earth, the distance fallen at the distance of the moon is 4.9/3600 meters = 1/20 inch! This is the theoretical prediction of Newton’s Gravitational Theory.

38. 1/20 inch? • How far does the moon actually fall in 1 second? v = d/t d is the circumference of its orbit = 2 p (60R) t is the orbital period ~ 1 month v is the moon's orbital speed = 2 p (60R)/(1 month)

39. Projectile Orbit Geometry line segment 0A = 0D = r = 60R line segment AC = d = v/t line segment CD = s r2 + d2 = (s+r)2 r2 + d2 = s2 + r2 + 2rs d2 = 2rs + s2 d2 = 2rs s = d2/2r

40. 1/20 inch s = d2/2r s = [(2 p 60R/1 month) 1 second]2/2(60R) 60R = 1.513x1010 inches 1 month = 27.32 days (sidereal) s ~ 1/20 inch

41. Newton’s Orbit Cannon • How much Energy is Required?

42. Orbits Object must have kinetic energy greater than the gravitational potential energy needed to escape the earth. The velocity associated with this kinetic energy is the escape velocity.

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