Dynamics of Circular Motion

Physics: Chapter 5

  1. Definitions
    1. Uniform Circular Motion: motion of an object traveling at a constant speed on a circular path
    2. Period (T): time need for one revolution of circle


  2. Speed (v)
    1. Circumference is distance (2pr)
    2. r measured at right angle to rotation
    3. Period (T) is time
    4. v = (2pr)/T
    5. unit: m/s


  3. Centripetal Acceleration (ac)
    1. Depends on v and r
    2. Use similar triangles to determine
    3. Dv/v = (vDt)/r
    4. Solve for acceleration Dv/Dt
    5. Magnitude: ac = v2/r
    6. units: m/s2
    7. Direction: toward center of circle, continuously changing


  4. Centripetal Force
    1. Magnitude: F = ma = mv2/r
    2. Direction: toward center of circle, continuously changing


  5. Banked Curves: Analysis:
    1. Fc determined by r of curve and v of car
    2. Fc also equal to component of FN toward center of curve
    3. That component is opposite q, so multiply FN by sine
    4. Vertical component of FN balances weight of car (mg) [(FN cos q) = mg]


  6. Bank Angle Solution
    1. Fc = FN sin q = mv2/r
    2. Divide by weight
    3. (FN sin q)/(FN cos q) = (mv2/r)/mg
    4. sin/cos = tan
    5. tan q = v2/rg


  7. The Motion of Satellites
    1. Kepler's First Law
      1. The orbit of each planet is an ellipse with the sun at one focus
      2. By determining the curve in one part we can deduce the whole orbit
    2. Kepler's Second Law
      1. An imaginary line from the sun to a moving planet sweeps out equal area in equal times.
      2. Predict speed of satellites for rendezvous
    3. Kepler's Third Law
      1. The ratio of the square of a planet's period to the cube of its average distance from the sun is a constant.
      2. Constant is the same for all things orbiting a body
      3. T2/R3 = k


  8. Satellites: Orbital Speed
    1. Fc of satellite:
    2. Fc is same as force of gravity: G(mME)/r2 (from Newton)
      1. (r is measure to center of Earth)
      2. m is mass of satellite
    3. Fc also is mv2/r
    4. Use these equations to solve for v
    5. Set equations equal to each other
    6. G(mME)/r2 = mv2/r
    7. Solve for v
    8. v = GME /r
    9. Only 1 speed is possible for given r
    10. Speed is independent of m


  9. GPS
    1. Satellites in geosychronous orbit
    2. v, G, ME known
    3. Clock in receiver synchronized with satellite
    4. Each satellite determines value for r
    5. Intersection of circles is location


  10. Orbital Period
    1. v = 2pr/T
    2. v is also v = GME /r
    3. Solve these equations for T
    4. T = 2pr3/2 / GME
    5. This is Kepler's 3rd law


  11. Apparent Weightlessness
    1. Apparent weight is force of one object pushing on another
    2. If both objects fall (are accelerated) at same speed apparent weight is 0
    3. Objects in orbit fall to center of Earth at same rate and appear weightless


  12. Artificial Gravity
    1. Rotation of space station creates Fc at outer surface of station
    2. Surface of station pushes person in station with Fc
    3. This push is perceived as gravity
    4. Results in apparent weight
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