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Rotational Kinematics

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If you spin a wheel, and look at how fast a point on the wheel is spinning, the ... gimbals which have nearly frictionless bearings and which isolate the central ... – PowerPoint PPT presentation

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Title: Rotational Kinematics


1
Rotational Kinematics
2
Rotation about a fixed axis
  • Rotational Motion
  • Remember a circle is just a straight line rolled
    up
  • If you spin a wheel, and look at how fast a point
    on the wheel is spinning, the answer depends on
    how far away the point is from the center. Linear
    velocity isn't the most convenient quantity to
    use when you're dealing with rotation, and for
    the same reason neither is displacement, or
    acceleration
  • it is often more convenient to use rotational
    equivalents.
  • The equivalent variables for rotation are
  • angular displacement (angle, for short)
  • angular velocity
  • angular acceleration
  • All the angular variables are related to the
    straight-line variables by a factor of r, the
    distance from the center of rotation to the point
    you're interested in.

I think my head is spinning
3
Rotation about a fixed axis
  • Rolling Motion
  • When an object such as a wheel or a ball rolls,
    it does not slip where it makes contact with the
    ground.
  • With a car or a bicycle tire, there is friction
    between the tire and the road, and if the tire is
    rolling then the frictional force is a static
    force of friction.
  • This is because there is no slipping, so the
    point on the tire in contact with the road is
    instantaneously at rest.
  • For a point on the outside of the tire, the
    rotational speed happens to be equal to the
    linear speed of the car this is because each
    time the tire makes a complete revolution, the
    car will have traveled a distance equal to the
    circumference of the tire, so the linear distance
    and the rotational distance are the same for the
    same time interval.

Now were rolling!
4
Rotational Dynamics
5
Rotational Dynamics
  • Torque
  • A torque is a force exerted at a distance from an
    axis of rotation the easiest way to think of
    torque is to consider a door.
  • When you open a door, where do you push?
  • If you exert a force at the hinge, the door will
    not move
  • the easiest way to open a door is to exert a
    force on the side of the door opposite the hinge,
    and to push or pull with a force perpendicular to
    the door. This maximizes the torque you exert.
  • We can state the equation for torque as

6
Rotational Dynamics
  • Center of Gravity
  • The center of gravity of an object is the point
    you can suspend the object from without there
    being any rotation because of the force of
    gravity, no matter how the object is oriented. If
    you suspend an object from any point, let it go
    and allow it to come to rest, the center of
    gravity will lie along a vertical line that
    passes through the point of suspension. Unless
    you've been exceedingly careful in balancing the
    object, the center of gravity will generally lie
    below the suspension point.
  • The center of gravity is an important point to
    know, because when you're solving problems
    involving large objects, or unusually-shaped
    objects, the weight can be considered to act at
    the center of gravity. In other words, for many
    purposes you can assume that object is a point
    with all its weight concentrated at one point,
    the center of gravity.
  • The center of mass of an object is generally the
    same as its center of gravity. Very large
    objects, large enough that the acceleration due
    to gravity varies in different parts of the
    object, are the only ones where the center of
    mass and center of gravity are in different
    places.
  • An object thrown through the air may spin and
    rotate, but its center of gravity will follow a
    smooth parabolic path, just like a ball.

7
Rotational Dynamics
  • Moment of Inertia (Rotational Mass)
  • the rotational equivalent of mass, which is
    something called the moment of inertia.
  • Mass is a measure of how difficult it is to get
    something to move in a straight line, or to
    change an object's straight-line motion. The more
    mass something has, the harder it is to start it
    moving, or to stop it once it starts. Similarly,
    the moment of inertia of an object is a measure
    of how difficult it is to start it spinning, or
    to alter an object's spinning motion. The moment
    of inertia depends on the mass of an object, but
    it also depends on how that mass is distributed
    relative to the axis of rotation an object where
    the mass is concentrated close to the axis of
    rotation is easier to spin than an object of
    identical mass with the mass concentrated far
    from the axis of rotation.
  • The moment of inertia of an object depends on
    where the axis of rotation is. The moment of
    inertia can be found by breaking up the object
    into little pieces, multiplying the mass of each
    little piece by the square of the distance it is
    from the axis of rotation, and adding all these
    products up
  • For common objects rotating about typical axes of
    rotation, these sums have been worked out, so we
    don't have to do it ourselves. A table of some of
    these moments of inertia can be found on the next
    slide

Divers minimizing their moments of inertia in
order to increase their rates of rotation.
8
Rotational Dynamics
  • Moment of Inertia

9
Rotational Dynamics
  • Angular Momentum
  • The angular momentum of a moving particle is
  • where m is its mass, v? is the component of its
    velocity vector perpendicular to the line joining
    it to the axis of rotation, and r is its distance
    from the axis. Positive and negative signs are
    used to describe opposite directions of rotation.
  • The angular momentum of a finite-sized object or
    a system of many objects is found by dividing it
    up into many small parts, applying the equation
    to each part, and adding to find the total amount
    of angular momentum.
  • Conservation of angular momentum has been
    verified over and over again by experiment, and
    is now believed to be one of the three most
    fundamental principles of physics, along with
    conservation of energy and momentum.

10
Rotational Dynamics
  • Angular Momentum Conserved
  • When a figure skater is twirling, there is very
    little friction between her and the ice, so she
    is essentially a closed system, and her angular
    momentum is conserved. If she pulls her arms in,
    she is decreasing r for all the atoms in her
    arms. It would violate conservation of angular
    momentum if she then continued rotating at the
    same speed, i.e., taking the same amount of time
    for each revolution, because her arms'
    contributions to her angular momentum would have
    decreased, and no other part of her would have
    increased its angular momentum. This is
    impossible because it would violate conservation
    of angular momentum. If her total angular
    momentum is to remain constant, the decrease in r
    for her arms must be compensated for by an
    overall increase in her rate of rotation. That
    is, by pulling her arms in, she substantially
    reduces the time for each rotation.

11
Rotational Dynamics
  • Rotational and Translational Analogies

12
Rotational Dynamics
  • Application of Conservation of Angular Momentum
    and Torque
  • Angular displacement, angular velocity, and
    angular acceleration are all vectors, too. But
    which way do they point? Every point on a rolling
    tire has the same angular velocity, and the only
    way to ensure that the direction of the angular
    velocity is the same for every point is to make
    the direction of the angular velocity
    perpendicular to the plane of the tire.
  • To figure out which way it points, use your right
    hand.
  • Stick your thumb out as if you're hitch-hiking,
    and curl your fingers in the direction of
    rotation. Your thumb points in the direction of
    the angular velocity.
  • If you look directly at something and it's
    spinning clockwise, the angular velocity is in
    the direction you're looking if it goes
    counter-clockwise, the angular velocity points
    towards you. Apply the same thinking to angular
    displacements and angular accelerations.

13
Rotational Dynamics
  • Application of Conservation of Angular Momentum
    and Torque
  • Gyroscopes
  • A gyroscope is a device for measuring or
    maintaining orientation, based on the principle
    of conservation of angular momentum. The essence
    of the device is a spinning wheel on an axle.
  • The classic image of a gyroscope is a fairly
    massive rotor suspended in light supporting rings
    called gimbals which have nearly frictionless
    bearings and which isolate the central rotor from
    outside torques. At high speeds, the gyroscope
    exhibits extraordinary stability of balance and
    maintains the direction of the high speed
    rotation axis of its central rotor. The
    implication of the conservation of angular
    momentum is that the angular momentum of the
    rotor maintains not only its magnitude, but also
    its direction in space in the absence of external
    torque. The classic type gyroscope finds
    application in gyro-compasses, but there are many
    more common examples of gyroscopic motion and
    stability. Spinning tops, the wheels of bicycles
    and motorcycles, the spin of the Earth in space,
    even the behavior of a boomerang are examples of
    gyroscopic motion.

Gyroscopes are used in inertial navigation
systems to navigate jetliners and ships across
the oceans
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