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Chapter 7

Chapter 7. Circular Motion and Gravitation. Angular Measure. Circular system of position measurement Useful when objects move in circular (or partially circular) paths Planetary motion (orbits) Useful when objects are rotating Planetary motion (rotation). Angular measure.

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Chapter 7

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  1. Chapter 7 Circular Motion and Gravitation

  2. Angular Measure • Circular system of position measurement • Useful when objects move in circular (or partially circular) paths • Planetary motion (orbits) • Useful when objects are rotating • Planetary motion (rotation)

  3. Angular measure • Rectangular (Cartesian) coordinates • x—perpendicular distance from the vertical axis • y—perpendicular distance from the horizontal axis • Angular (circular) coordinates • If a line is drawn directly from the origin to any point . . . • r—distance from the origin • --angle made with the horizontal • Units of degrees or radians • x = rcos • y = rsin

  4. Angular measure • Displacement • x or y = linear displacement •  = angular displacement • For a circle • r = 0, so only angular displacement is needed to describe motion

  5. Angular measure • Radians • s = arclength • Distance around a circular path • 1 radian = angle that produces an arclength equal to r • 1 rad = 57.3 • Degrees • 1 revolution = 360 = 2 rad • 1 = 60 minutes • 1 minute = 60 seconds • *parsec

  6. Angular measure • Angular speed () • Units of rad/s • rpm—revolutions per minute • Common unit, but not standard metric • Angular velocity () • Direction given by right hand rule • Curl fingers in direction of rotation () • Thumb points in direction of 

  7. Angular measure • Instantaneous linear velocity (v) is in a straight line tangent to the circular path • v = r • For constant angular velocity, tangential (linear) velocity increases with the distance from the origin • How CD’s work

  8. Angular measure • Period (T)—time required for 1 complete revolution (cycle) • Units of seconds (s) • Frequency (f)—number of revolutions completed every second • Units of 1/s, s-1, cycles/s • Renamed Hertz (Hz)

  9. Uniform Circular Motion • Uniform circular motion—object moving at a constant speed in a circle • Speed is constant, but velocity is not • Direction is constantly changing • Since velocity is changing, acceleration must be present

  10. Uniform Circular Motion

  11. Uniform Circular Motion

  12. Uniform Circular Motion • Instantaneous acceleration points toward the center of the circle • Centripetal acceleration—”center seeking” acceleration • Caused uniform circular motion • Instantaneous velocity vector is tangent to the circle Example 7.5, 7.6, page 219

  13. Uniform Circular Motion • Centripetal Force—”center seeking” force • Net force needed for acceleration • Any force that applies a centripetal acceleration is a centripetal force • Can be friction, gravity, tension of a string • Centripetal force does no work Example 7.8, p. 223 Centripetal F and a for the Moon

  14. Angular Kinematics • Angular acceleration () = rate of change of angular velocity • Does not apply to Uniform Circular Motion • Related to linear (tangential) acceleration Example 7.10, p. 225

  15. Angular Kinematics • Angular Kinematic Equations Example 7.11, p. 226

  16. Gravitation • Newton’s Law of Gravitation • m1 = mass of object 1 • m2 = mass of object 2 • r = distance (center to center) between objects • G = 6.67 x 10-11 Nm2/kg2 • Universal Gravitational Constant Acceleration due to gravity FG and aG Moon-Earth

  17. Kepler’s Laws • Kepler’s 1st Law (Law of orbits) • Planets move in elliptical paths with the Sun at one of the focal points

  18. Kepler’s Laws

  19. Kepler’s Laws • Kepler’s 2nd Law (Law of areas) • A line from the sun to a planet sweeps out equal areas in equal times

  20. Kepler’s Laws • Kepler’s 3rd Law • Square of the orbital period is directly proportional to the cube of the average distance from the planet to the Sun • T = orbital period • r = average distance from the sun • K = 2.97 x 10-19 s2/m3 (for objects orbiting our sun) Geosynchronous satellite Newton’s Derivation

  21. Escape Velocity • vesc = escape velocity • Tangential velocity needed to escape from the surface of a planet (or any other object) • M = Mass of the planet • R = Radius of the planet

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