1. Uniform Circular Motion Physics 12

2. Centripetal Acceleration In order for an object to follow a circular path, a force needs to be applied in order to accelerate the object Although the magnitude of the velocity may remain constant, the direction of the velocity will be constantly changing As a result, this force will provide a centripetal acceleration towards the centre of the circular path

4. How can we calculate centripetal acceleration?

5. Centripetal Force Like the centripetal acceleration, the centripetal force is always directed towards the centre of the circle The centripetal force can be calculated using Newton’s Second Law of Motion

6. Problem – horizontal circle A student attempts to spin a rubber stopper (m = 0.050kg) in a horizontal circle with a radius of 0.75m. If the stopper completes 2.5 revolutions every second, determine the following: –The centripetal acceleration –The centripetal force

7. The stopper will cover a distance that is 2.5 times the circumference of the circle every second Determine the circumference Multiply by 2.5 Use the distance and time (one second) to calculate the speed of the stopper

8. Use the speed and radius to determine the centripetal acceleration Then use the centripetal acceleration and mass to determine the centripetal force

9. Problem – vertical circle A student is on a carnival ride that spins in a vertical circle. –Determine the minimum speed that the ride must travel in order to keep the student safe if the radius of the ride is 3.5m. –Determine the maximum force the student experiences during the ride (in terms of number of times the gravitational force)

10. Problem – vertical circle

11. Vertical Circle While travelling in a vertical circle, gravity must be considered in the solution While at the top of the circle, gravity acts towards the centre of the circle and provides some of the centripetal force While at the bottom of the circle, gravity acts away from the centre of the circle and the force applied to the object must overcome both gravity and provide the centripetal force

12. Vertical Circle To determine the minimum velocity required, use the centripetal force equal to the gravitational force (as any slower than this and the student would fall to the ground) To determine the maximum force the student experiences, consider the bottom of the ride when gravity must be overcome

13. At the top of the circle, set the gravitational force (weight) equal to the centripetal force Solve for velocity

14. At the bottom of the circle, the net force is equal to the sum of the gravitational force and the centripetal force Solve for number of times the acceleration due to gravity

15. Road Design You are responsible to determine the speed limit for a turn on the highway. The radius of the turn is 55m and the coefficient of static friction between the tires and the road is 0.90. –Find the maximum speed at which a vehicle can safely navigate the turn –If the road is wet and the coefficient drops to 0.50, how does this change the maximum speed

16. Diagrams

17. The maximum speed at which a vehicle can safely navigate the turn

18. Coefficient drops to 0.50, how does this change the maximum speed

19. The Motion of Planets Birth of Modern Astronomy OR How Nerds Changed the World!!!

20.  explain qualitatively Kepler’s first and second laws and apply quantitatively Kepler’s third law explain and apply the law of universal gravitation to orbital notations by using appropriate numeric and graphic analysis  distinguish between scientific questions and technological problems as applied to orbital situations Learning Outcomes (Students will be able to…):

21. Assumptions of Early Models of the Solar System (from the time of Aristotle…) Geocentric - Earth in the middle Everything orbits the Earth Stars are located on the Celestial Sphere Everything moves in uniform circular motions

22. Earth Deferent Epicycle Mars Equant Claudius Ptolemy (87-165)

23. Nicolaus Copernicus (1473-1543) Errors building up Must be a better way! Let’s try a Heliocentric (or Sun- centered) system! Not any better though

25. Tycho Brahe (1546-1601) Comet – beyond the Moon Supernova – far away Naked eye observations of planets Accuracy through repetition Best observations of planetary positions Hired “nerd” to help calculate model Died….

26. Johannes Kepler (1571-1630) Worked for Brahe Took data after his death Spent years figuring out the motions of the planets Came up with… Three Laws of Planetary Motion

27. 1 st Law: Planets move in elliptical orbits with the Sun at one foci Sun Foci (sing. Focus) PerihelionAphelion Average distance from the Sun = 1 Astronomical Unit (1 A.U.) = approx. 150 000 000 km

28. 2 nd Law: Planets move faster at perihelion than at aphelion OR a planet sweeps out equal areas in equal time periods. 1 Month

29. 3 rd Law: Period is related to average distance T = period of the orbit r = average distance kT 2 = r 3 Longer orbits - greater average distance Need the value of k to use the formula k depends upon the situation Can be used for anything orbiting anything else

30. Special version of Kepler’s third Law – If the object is orbiting the Sun T – measured in years, r – measured in A. U., then…. T 2 = r 3

31. For planets A and B, Kepler’s 3 rd Law can look like this…

32. Galileo Galilei (1564-1642) Knew of Copernicus’s & Kepler’s work Used a telescope to look at the sky What did he see?

33. The Moon was an imperfect object Venus has phases

34. Jupiter has objects around it Saturn is imperfect The Sun is imperfect

35. Isaac Newton (1642-1727) The ultimate “nerd” Able to explain Kepler’s laws Had to start with the basics - The Three Laws of Motion

36. 1. Law of Inertia - Objects do whatever they are currently doing unless something messes around with them.

37. 2. Force defined F = ma F=force m=mass a=acceleration (change in motion)

38. 3. For every action there is an equal and opposite reaction. The three laws of motion form the basis for the most important law of all (astronomically speaking) Newton’s Universal Law of Gravitation

39. F=force of gravity G=constant (6.67 x 10 -11 Nm 2 /kg 2 ) M 1, M 2 = masses R=distance from “centers” Gravity is the most important force in the Universe

40. An Inverse Square Law…

41. Newton’s Revisions to Kepler’s Laws Newton agreed with 1 st law of motion Defined bound orbits (i.e. circular, elliptical) and unbound orbits (i.e. hyperbolic, parabolic) with Sun at one focus Used conic sections to describe orbits

42. Newton’s Revisions to Kepler’s Laws Newton agreed with 2 nd law of motion Believed planetary motion to be non- constant acceleration due to varying distance between planet and Sun Force causing acceleration was gravity

43. 4π2 and G are just constant #s (they don’t change) M 1 and M 2 are any two celestial bodies (could be a planet and Sun) Importance: if you know period and average distance of a planet, you can find mass of Sun (2 x 10 30 kg) or any planet! Mass of Sun is 2 000 000 000 000 000 000 000 000 000 000 kg Mass of Earth is 6 000 000 000 000 000 000 000 000 kg Mass of Mr. J is 100 kg! WOW! Newton’s Revisions to Kepler’s Laws Newton extended 3 rd law to…

44. Newton’s Mountain Horizontal projectile launched at 8km/s How far does the projectile fall in one second? How far does the Earth “fall” away from the projectile? –Assume that arc length and chord length are equal over the 8km distance and the Earth’s radius is 6400km

45. Newton’s Mountain Shortly after developing the Universal Law of Gravitation, Newton began a series of thought experiment involving artificial satellites Newton’s thought was that if you had a tall enough mountain and launched a cannonball fast enough horizontally, it would fall towards the Earth at the same rate the Earth would “fall” away This would result in the cannonball orbiting the Earth

47. Geostationary Satellite A geostationary (or geosynchronous) satellite, will always be above the same spot on Earth What is the orbital radius, altitude and speed of a geostationary satellite? –Use Newton’s Version of Kepler’s Law to solve for orbital radius –Subtract Earth’s radius from orbital radius to determine altitude –Set gravitational force equal to centripetal force and solve for orbital speed

48. Weightlessness

49. The International Space Station orbits at an altitude of 226km; determine the force of gravity on an astronaut (65.kg) at this altitude and compare this to their weight on the surface of the Earth 594N at the ISS 638N on Earth Is the astronaut “weightless?”

50. Weightlessness Weightlessness occurs because objects are all falling towards the surface of the Earth at the same rate NASA simulates this on the “Vomit Comet” a high altitude aircraft that plunges toward Earth

51. Another way to look at “g”…

53. Another way to look at gravitational potential energy of an object… (h is height but since it is arbitrary, it can be chosen as the distance from the center of the Earth to the position of the object…or r)

54. Some important orbital applications… Geosynchronous means having an orbit around the Earth with a period of 24 hours

55. Gravitational Field The strength of a gravitational field can be determined using a test mass (m t ) The mass should be very small compared to the mass creating the field A gravitational field will be measured in the units of N/kg

56. Gravitational Field – One Mass Arrows point toward the mass showing the direction of force experienced by a mass at that point Circles represent equal potential (i.e. where the force experienced by a mass would be equivalent

57. Gravitational Field – Two Masses Note that earth’s field is much larger than the moon’s Small changes in the field as you approach the moon

58. Motion of a Particle in a Gravitational Field While the gravitational field lines show the direction of the force, they do not indicate the direction of motion Consider the behaviour of our solar system –http://janus.astro.umd.edu/javadir/orbits/ssv.ht mlhttp://janus.astro.umd.edu/javadir/orbits/ssv.ht ml

59. Spacetime One of Einstein’s contributions is the concept of spacetime –In spacetime, space and time are warped by massive objects –The resulting curved spacetime results in objects following a curved path as they move through spacetime –This is easily modeled by considering massive objects on a flexible sheet

60. Einstein viewed gravity and the motion of celestial objects, like planets, VERY differently… (1875 – 1955) Curved space-time effects both mass and light!

61. Black Holes

62. In 1916, Einstein published his theory of general relativity (GR), which discussed gravity and explained how the presence of matter causes space and time to be warped.

63. Light travel in curved spacetime Photons of light passing near our Sun will move the same way through curved space. They will “bend” around the Sun.

64. Time runs slower in curved spacetime A fundamental tenet of GR is that time runs more slowly in curved spacetime.

65. The Principle of Equivalence, which states: All local, freely falling, non-rotating laboratories are fully equivalent for the performance of all physical experiments.

66. Gravitational Redshift Another important effect of Einstein’s theory of GR is gravitational redshift  photons lose energy as they try to escape from a strong gravitational field. energy of a photon is inversely proportional to its wavelength. Characteristic of systems containing high-density objects such as neutron stars and white dwarfs, although the effect would be particularly strong if the system contained a black hole.

67. The Schwarzschild radius, Rs, is given by

68. Any star that collapses beyond its Schwarzschild radius is called a black hole. coined by the American mathematical physicist John Wheeler in 1968. A black hole is enclosed by an event horizon, the surface of which is described by a sphere with r = R S. At the centre of the event horizon is a singularity, a point of zero volume and infinite density where all of the black hole’s mass is located.

69. Dark Matter