10'4 Rotational Kinetic Energy

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10.4 Rotational Kinetic Energy

• An object rotating about some axis with an
  angular speed, ?, has rotational kinetic energy
  even though it may not have any translational
  kinetic energy
• Each particle has a kinetic energy of
• Ki 1/2 mivi2
• Since the tangential velocity depends on the
  distance, r, from the axis of rotation, we can
  substitute vi wi r

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Fig 10.6
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Rotational Kinetic Energy, cont

• The total rotational kinetic energy of the rigid
  object is the sum of the energies of all its
  particles
• Where I is called the moment of inertia

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Rotational Kinetic Energy, final

• There is an analogy between the kinetic energies
  associated with linear motion (K 1/2 mv 2) and
  the kinetic energy associated with rotational
  motion (KR 1/2 Iw2)
• Rotational kinetic energy is not a new type of
  energy, the form is different because it is
  applied to a rotating object
• The units of rotational kinetic energy are Joules
  (J)

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Moment of Inertia

• The definition of moment of inertia is
• The dimensions of moment of inertia are ML2 and
  its SI units are kg.m2
• We can calculate the moment of inertia of an
  object more easily by assuming it is divided into
  many small volume elements, each of mass Dmi

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Moment of Inertia, cont

• We can rewrite the expression for I in terms of
  Dm
• With the small volume segment assumption,
• If r is constant, the integral can be evaluated
  with known geometry, otherwise its variation with
  position must be known

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Moment of Inertia of a Uniform Solid Cylinder

• Divide the cylinder into concentric shells with
  radius r, thickness dr and length L
• Then for I

Fig 10.8
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Fig 10.7
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10. 5 Torque

• The moment arm, d, is the perpendicular distance
  from the axis of rotation to a line drawn along
  the direction of the force
• d r sin ?

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10.5 Definition of Torque

• Torque, t, is the tendency of a force to rotate
  an object about some axis
• Torque is a vector
• t r F sin f F d
• F is the force
• f is the angle between the force and the
  horizontal (the line from the axis to the
  position of the force)
• d is the moment arm (or lever arm)

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Torque, cont.

• The horizontal component of the force (F cos f)
  has no tendency to produce a rotation
• Torque will have direction
• If the turning tendency of the force is
  counterclockwise, the torque will be positive
• If the turning tendency is clockwise, the torque
  will be negative

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Torque Unit

• The SI unit of torque is N.m
• Although torque is a force multiplied by a
  distance, it is very different from work and
  energy
• The units for torque are reported in N.m and not
  changed to Joules

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Torque as a Vector Product

• Torque is the vector product or cross product of
  two other vectors

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Vector Product, General

• Given any two vectors,
  and
• The vector product
• is defined as a third vector,
  whose magnitude is
• The direction of C is given by the right-hand rule

Fig 10.13
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Properties of Vector Product

• The vector product is not commutative
• If is parallel (q 0o or 180o) to
  then
• This means that
• If is perpendicular to then

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Vector Products of Unit Vectors

• The signs are interchangeable
• For example,

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Net Torque on an object

• The force F1 will tend to cause a
  counterclockwise rotation about O
• The force F2 will tend to cause a clockwise
  rotation about O
• tnet t1 t2 F1d1 F2d2

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10.6 Force vs. Torque
Forces can cause a change in linear motion, which
is described by Newtons Second Law F Ma.
Torque can cause a change in rotational motion,
which is described by the equation t I a.
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The Rigid Object In Equilibrium

• The net external force must be equal zero
• The net external torque about any axis must be
  equal zero

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Fig 10.16(b) (c)
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10.7 Rotational motion of a rigid object under a
net torque

• The magnitude of the torque produced by a force
  around the center of the circle is
• t Ft r (mat) r
• The tangential acceleration is related to the
  angular acceleration
• St S(mat) r S(mra) r S(mr 2) a
• Since mr 2 is the moment of inertia of the
  particle,
• St Ia
• The torque is directly proportional to the
  angular acceleration and the constant of
  proportionality is the moment of inertia

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Fig 10.18(a) (b)
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Work in Rotational Motion

• Find the work done by a force on the object as it
  rotates through an infinitesimal distance ds r
  dq
• The radial component of the force does no work
  because it is perpendicular to the displacement

Fig 10.19
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Work in Rotational Motion, cont

• Work is also related to rotational kinetic
  energy
• This is the same mathematical form as the
  work-kinetic energy theorem for translation
• If an object is both rotating and translating, W
  DK DKR

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Power in Rotational Motion

• The rate at which work is being done in a time
  interval dt is the power
• This is analogous to P Fv in a linear system

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