CHAPTER 7 Circular Motion and the Laws of Gravity by ewghwehws


									              CHAPTER 7
Rotational Motion and the Law of Gravity
Angular Speed and Angular Acceleration

                 s = arc length
                 Θ = arc angle (radians)
                 Θ = s ; where r = radius
             Term              Symbol      Units
     Angular Displacement        ∆Θ       radians
       Angular Speed             ω      radians/sec
   Average Angular Speed         ω      radians/sec
(Average) Angular Acceleration         radians/sec2
   Θ2 – Θ1
   t2 – t1
     ∆Θ              Average Angular Speed
  ω=              Average rate at which the arc
                       angle is changing
              (Instantaneous) Angular Speed
ω = lim ∆Θ
             The rate at which the arc angle is
   ∆t0 ∆t changing at a particular instant in time
    ω2 – ω1
    t2 – t1
                  Average Angular Acceleration
   =         The average rate at which the angular
      ∆t               speed is changing

      NOTE: You are not required to work with
          changing rates of acceleration.
                 Therefore,  = 
         Linear/ Angular Comparison

     Term                Linear           Angular
 Displacement           ∆x m             ∆Θ   rad
    Velocity            v    m/s         ω rad/sec
 Acceleration           a    m/s2         rad/sec2

Formulas:       ∆x = ½(vo+v)t       ∆Θ = ½(ωo+ω)t
             ∆x = vot + ½at2        ∆Θ = ωot + ½ t2
               v = vo + at           ω = ωo + t
                v2 = vo2 + 2a∆x     ω2 = ωo2 + 2∆Θ
                      N/A               s = rΘ

      NOTE: All points on a rotating disc have the
              same values for ω and .
Example Problem (Rotating Disc)
A disc of radius 2.50m accelerates to an angular velocity of
1.35 radians/sec in 12.5 seconds. Calculate (a) the disc’s
angular acceleration; (b) the number of revolutions
completed during the acceleration; (c) the distance traveled
by a point on the edge of the disc.
Identify the variables
  ωo = 0.00 rad/sec        = ?
  ω = 1.35 rad/sec         t = 12.5 sec       ∆Θ = ?
Choose Appropriate Formulas
  ω = ωo + t  = .108 rad/sec2

  ∆Θ =
        [ ]ωo + ω
                    t    ∆Θ = ωot + ½ t2
                           ∆Θ = 8.44 rad
    ∆Θ = 8.44 rad
  1 revolution = 2 rad = 6.2 rad
    8.44 rad 1 rev
               6.28 rad                   ∆s = r∆Θ

      # rev = 1.34rev                    ∆s = 21.1m
Tangential Speed (vt)     •       arc length of the movement
                               time to complete the movement
                          • vt = ∆s
                            ∆s = r ∆Θ
                            vt = r ∆Θ
                                vt = r ω

Tangential Acceleration (t)         at = r 

    NOTE: Not every point on a rotating disc has the
    same tangential speed or tangential acceleration.
Centripetal Acceleration (center-seeking)

      The tangential velocity vt illustrated above is
    clearly changing. How can you tell it is changing?
      Its direction is changing.
         a = ∆v = vf – vi = vf + -vi
                ∆t        ∆t           ∆t
    The illustration on the right clearly shows that ∆v
    points towards the center of rotation.
Centripetal Acceleration (ac)
     ∆v = ∆s       from similar triangles
      v       r
     ∆v = v
              ∆s     and ∆v = ac ∆t
     ac ∆t =      ∆s
             v ∆s      and ∆s = v
     ac =
             r ∆t          ∆t
       ac =    = Centripetal Acceleration
Centripetal Force: • The net force that causes an object
                   to accelerate towards its center of
                   rotation (i.e. go in a circle)
                    • Fc = mac
                    • Fc is not a new force
                    • Fc will be either one or a
                    combination of more than one of the
                    following forces:
                           Fa (T)
Example Problem (Vertical loop)
A rollercoaster contains a loop 20m in diameter. a) What speed
will cause a passenger to feel “weightless” at the top of the
loop? b) If this same person has a mass of 90kg what weight
will they experience at the bottom of the loop?
                   Sketch the Force Diagrams

                                  Analysis: The only forces
Analysis: The person is           acting on the person are the
“weightless” if no forces other   normal force and force of
than gravity are acting upon      gravity.
them (i.e. FN=0).                   Fc = Fnet(center) = FN – Fg
Hence, Fg is Fc                     FN = Fc + Fg
 mg = mv2                               = mv2 + mg
       r                                    r
 v = √rg        v = 9.9m/s            FN = 1764 Newtons
Example Problem (Banked Curve)
a) For a car traveling with a speed v around a curve of
radius r, determine the formula for the angle at which
the road should be bank so that no friction is required.
b) what is this angle for an expressway off-ramp of
radius 50m at a design speed of 50km/hr?
Sketch the Force Diagram
                            Part a:
                               • Since there is no friction,
                               there is no component of
                               friction directed to the
                               center (right).
                               • Only the component
                               FNsinΘ of the normal force
                               is directed to the center.
                               • FN sinΘ is the net force
                               towards the center = Fc
                FN sinΘ = Fc
                FN sinΘ = mv2
                mg = FN cosΘ
                  mg · sinΘ = mv2
                 cosΘ           r
                  mg tanΘ = mv2

Part b:
                      tanΘ = v2
                             gr Θ = tan

   r = 50m
   v = 50km 1000m  1hr
                        = 14m/s
          hr   km 3600s
   g = 9.8
     Θ = 22°
      Newton’s Universal Law of Gravitation
 Every particle in the universe attracts every other particle
 with a force that is directly proportional to the product of
 their masses and inversely proportional to the square of
 the distance between them.

     Fg = G m1 m2

G = universal gravitational
G = 6.67 x 10-11 N·m2/kg2
Force of Earth’s Gravity (on Earth’s Surface)
Fg = G mearth mobject

Fg =
       [ ]
       G mearth
        rearth 2
                 · massobject

     (6.67 x10-11 N·m2/kg2)(5.98x1024 kg)
Fg =                                      · massobject
                 (6.37x10 6 m)2

Fg = 9.8 x massobject

    Fg = mobect g
          Velocity of an Earth Satellite
      What is the only Force acting on the satellite?
    If satellite is in circular orbit, then…
        Fg = Fc
        mg = mvs2               vs = velocity of satellite
        vs = √gr        g = GMearth
        vs =
              √     GMearth · r

            vs =     GMearth
          Gravitational Potential Energy
         PE = -GMEm         r = distance of object from
                  r                Earth’s center
          • from Calculus
          • PE = zero at infinite distance from Earth
          • PE = some negative value closer to the Earth
Escape Speed
• Speed at surface of Earth where the object’s KE1 + PE1 is
sufficient to reach infinite distance from Earth (PE2=0J)
with a final speed of 0m/s (KE2=0J)
• From Conservation of Energy:
   KE1 + PE1 = KE2 + PE1
   mv2 + -GMearthm

                                             [      ]
   2       Rearth                            2GMearth   1/2
   mv2 + -GMearthm               vescape =    Rearth
   2       Rearth

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