Chapters 10, 11

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Chapters 10, 11 Rotation and angular momentum
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• Rotation of a rigid body
• We consider rotational motion of a rigid body
  about a fixed axis
• Rigid body rotates with all its parts locked
  together and without any change in its shape
• Fixed axis it does not move during the rotation
• This axis is called axis of rotation
• Reference line is introduced

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• Angular position
• Reference line is fixed in the body, is
  perpendicular to the rotation axis, intersects
  the rotation axis, and rotates with the body
• Angular position the angle (in radians or
  degrees) of the reference line relative to a
  fixed direction (zero angular position)

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• Angular displacement
• Angular displacement the change in angular
  position.
• Angular displacement is considered positive in
  the CCW direction and holds for the rigid body as
  a whole and every part within that body

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• Angular velocity
• Average angular velocity
• Instantaneous angular velocity the rate of
  change in angular position

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• Angular acceleration
• Average angular acceleration
• Instantaneous angular acceleration the rate of
  change in angular velocity

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• Rotation with constant angular acceleration
• Similarly to the case of 1D motion with a
  constant acceleration we can derive a set of
  formulas

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• Relating the linear and angular variables
  position
• For a point on a reference line at a distance r
  from the rotation axis
• ? is measured in radians

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• Relating the linear and angular variables speed
• ? is measured in rad/s
• Period

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• Relating the linear and angular variables
  acceleration
• a is measured in rad/s2
• Centripetal acceleration

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• Rotational kinetic energy
• We consider a system of particles participating
  in rotational motion
• Kinetic energy of this system is
• Then

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• Moment of inertia
• From the previous slide
• Defining moment of inertia (rotational inertia)
  as
• We obtain for rotational kinetic energy

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• Moment of inertia rigid body
• For a rigid body with volume V and density ?(V)
  we generalize the definition of a rotational
  inertia
• This integral can be calculated for different
  shapes and density distributions
• For a constant density and the rotation axis
  going through the center of mass the rotational
  inertia for 8 common body shapes is given in
  Table 10-2 (next slide)

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Moment of inertia rigid body
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• Moment of inertia rigid body
• The rotational inertia of a rigid body depends
  on the position and orientation of the axis of
  rotation relative to the body

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Chapter 10 Problem 25
Four equal masses m are located at the corners of
a square of side L, connected by essentially
massless rods. Find the rotational inertia of
this system about an axis (a) that coincides with
one side and (b) that bisects two opposite sides.
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• Parallel-axis theorem
• Rotational inertia of a rigid body with the
  rotation axis, which is perpendicular to the xy
  plane and going through point P
• Let us choose a reference frame, in which the
  center of mass coincides with the origin

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Parallel-axis theorem
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Parallel-axis theorem
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Parallel-axis theorem
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Chapter 10 Problem 51
A uniform rectangular flat plate has mass M and
dimensions a by b. Use the parallel-axis theorem
in conjunction with Table 10.2 to show that its
rotational inertia about the side of length b is
Ma2/3.
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• Torque
• We apply a force at point P to a rigid body that
  is free to rotate about an axis passing through O
• Only the tangential component Ft F sin f of
  the force will be able to cause rotation

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• Torque
• The ability to rotate will also depend on how
  far from the rotation axis the force is applied
• Torque (turning action of a force)
• SI unit Nm (dont confuse with J)

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• Torque
• Torque
• Moment arm r- r sinf
• Torque can be redefined as
• force times moment arm
• t F r-

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• Newtons Second Law for rotation
• Consider a particle rotating under the influence
  of a force
• For tangential components
• Similar derivation for rigid body

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Newtons Second Law for rotation
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Chapter 10 Problem 57
A 2.4-kg block rests on a slope and is attached
by a string of negligible mass to a solid drum of
mass 0.85 kg and radius 5.0 cm, as shown in the
figure. When released, the block accelerates down
the slope at 1.6 m/s2. Find the coefficient of
friction between block and slope.
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• Rotational work
• Work
• Power
• Work kinetic energy theorem

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Corresponding relations for translational and
rotational motion
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• Smooth rolling
• Smooth rolling object is rolling without
  slipping or bouncing on the surface
• Center of mass is moving at speed vCM
• Point of momentary contact between the two
  surfaces is moving at speed vCM
• s ?R
• ds/dt d(?R)/dt R d?/dt
• vCM ds/dt ?R

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• Rolling translation and rotation combined
• Rotation all points on the wheel move with the
  same angular speed ?
• Translation all point on the wheel move with
  the same linear speed vCM

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Rolling translation and rotation combined
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• Rolling pure rotation
• Rolling can be viewed as a pure rotation around
  the axis P moving with the linear speed vcom
• The speed of the top of the rolling wheel will
  be
• vtop (?)(2R)
• 2(?R) 2vCM

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• Friction and rolling
• Smooth rolling is an idealized mathematical
  description of a complicated process
• In a uniform smooth rolling, P is at rest, so
  theres no tendency to slide and hence no
  friction force
• In case of an accelerated smooth rolling
• aCM a R
• fs opposes tendency to slide

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Rolling down a ramp
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Chapter 10 Problem 39
What fraction of a solid disks kinetic energy is
rotational if its rolling without slipping?
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• Vector product of two vectors
• The result of the vector (cross) multiplication
  of two vectors is a vector
• The magnitude of this vector is
• Angle f is the smaller of the two angles between
  and

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• Vector product of two vectors
• Vector is perpendicular to the plane that
  contains vectors and and its direction is
  determined by the right-hand rule
• Because of the right-hand rule, the order of
  multiplication is important (commutative law does
  not apply)
• For unit vectors

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Vector product in unit vector notation
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• Torque revisited
• Using vector product, we can redefine torque
  (vector) as

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• Angular momentum
• Angular momentum of a particle of mass m and
  velocity with respect to the origin O is
  defined as
• SI unit kgm2/s

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Newtons Second Law in angular form
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Angular momentum of a system of particles
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• Angular momentum of a rigid body
• A rigid body (a collection of elementary masses
  ?mi) rotates about a fixed axis with constant
  angular speed ?
• For sufficiently symmetric objects

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• Conservation of angular momentum
• From the Newtons Second Law
• If the net torque acting on a system is zero,
  then
• If no net external torque acts on a system of
  particles, the total angular momentum of the
  system is conserved (constant)
• This rule applies independently to all components

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Conservation of angular momentum
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Conservation of angular momentum
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More corresponding relations for translational
and rotational motion
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Chapter 11 Problem 28
A skater has rotational inertia 4.2 kgm2 with
his fists held to his chest and 5.7 kgm2 with
his arms outstretched. The skater is spinning at
3.0 rev/s while holding a 2.5-kg weight in each
outstretched hand the weights are 76 cm from his
rotation axis. If he pulls his hands in to his
chest, so theyre essentially on his rotation
axis, how fast will he be spinning?
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Questions?
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Answers to the even-numbered problems Chapter 10
Problem 24 0.072 Nm
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Answers to the even-numbered problems Chapter 10
Problem 30 2.58 1019 Nm
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Answers to the even-numbered problems Chapter 10
Problem 40 hollow
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Answers to the even-numbered problems Chapter 11
Problem 16 69 rad/s 19 west of north
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Answers to the even-numbered problems Chapter 11
Problem 18 (a) 8.1 Nm kˆ (b) 15 Nm kˆ
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Answers to the even-numbered problems Chapter 11
Problem 24 1.7 10-2 Js
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Answers to the even-numbered problems Chapter 11
Problem 26 (a) 1.09 rad/s (b) 386 J
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Answers to the even-numbered problems Chapter 11
Problem 30 along the x-axis or 120 clockwise
from the x-axis
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Answers to the even-numbered problems Chapter 11
Problem 42 26.6