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6.1 Rotation Angle and Angular Velocity

  • Uniform circular motion is motion in a circle at constant speed. The rotation angle ΔθΔθ is defined as the ratio of the arc length to the radius of curvature:

    where arc length ΔsΔs is distance traveled along a circular path and rr is the radius of curvature of the circular path. The quantity ΔθΔθ is measured in units of radians (rad), for which

    rad=360º1 revolution.rad=360º1 revolution.
  • The conversion between radians and degrees is 1rad=57.3º1rad=57.3º.
  • Angular velocity ωω is the rate of change of an angle,

    where a rotation ΔθΔθ takes place in a time ΔtΔt. The units of angular velocity are radians per second (rad/s). Linear velocity vv and angular velocity ωω are related by

    v= or ω=vr.v= or ω=vr.

6.2 Centripetal Acceleration

  • Centripetal acceleration acac is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. It is perpendicular to the linear velocity vv and has the magnitude
    a c = v 2 r ; a c = 2 . a c = v 2 r ; a c = 2 .
  • The unit of centripetal acceleration is m/s2m/s2.

6.3 Centripetal Force

  • Centripetal force FcFc is any force causing uniform circular motion. It is a “center-seeking” force that always points toward the center of rotation. It is perpendicular to linear velocity vv and has magnitude

    which can also be expressed as

    F c = m v 2 r or F c = mr ω 2 , F c = m v 2 r or F c = mr ω 2 ,

6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force

  • Rotating and accelerated frames of reference are non-inertial.
  • Fictitious forces, such as the Coriolis force, are needed to explain motion in such frames.

6.5 Newton’s Universal Law of Gravitation

  • Newton’s universal law of gravitation: Every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. In equation form, this is

    where F is the magnitude of the gravitational force. GG is the gravitational constant, given by G=6.674×10–11Nm2/kg2G=6.674×10–11Nm2/kg2.

  • Newton’s law of gravitation applies universally.

6.6 Satellites and Kepler’s Laws: An Argument for Simplicity

  • Kepler’s laws are stated for a small mass mm orbiting a larger mass MM in near-isolation. Kepler’s laws of planetary motion are then as follows:

    Kepler’s first law

    The orbit of each planet about the Sun is an ellipse with the Sun at one focus.

    Kepler’s second law

    Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times.

    Kepler’s third law

    The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun:

    T1 2T2 2=r1 3r2 3,T1 2T2 2=r1 3r2 3,

    where TT is the period (time for one orbit) and rr is the average radius of the orbit.

  • The period and radius of a satellite’s orbit about a larger body MM are related by
    T 2 = 2 GM r 3 T 2 = 2 GM r 3


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