Rotation of a Rigid Object About a Fixed Axis

1
Chapter 10

• Rotation of a Rigid Object About a Fixed Axis

2
Rolling Motion

• Rigid bodyobject or system of particles in which
  distances between component parts remains
  constant
• Translational motionmovement of the system as a
  whole
• Described by the motion of the center of mass
• Rotational motionmovement of individual parts
  around a particular axis
• The spinning

3
Angular Measure

• Circular system of position measurement
• Useful when objects move in circular (or
  partially circular) paths
• Planetary motion (orbits)
• Useful when objects are rotating
• Planetary motion (rotation)

4
Angular measure

• Rectangular (Cartesian) coordinates
• xperpendicular distance from the vertical axis
• yperpendicular distance from the horizontal axis
• Angular (circular) coordinates
• If a line is drawn directly from the origin to
  any point . . .
• rdistance from the origin
• ?--angle made with the horizontal
• Units of degrees or radians
• x rcos?
• y rsin?

5
Angular measure

• Displacement
• ?x or ?y linear displacement
• ?? angular displacement
• For a circle
• ?r 0, so only angular displacement is needed to
  describe motion

6
Angular measure

• Radians
• s arclength
• Distance around a circular path
• 1 radian angle that produces an arclength equal
  to r
• 1 rad 57.3?
• Degrees
• 1 revolution 360? 2? rad
• 1? 60 minutes
• 1 minute 60 seconds
• parsec

7
Angular measure

• Angular speed (?)
• Units of rad/s
• rpmrevolutions per minute
• Common unit, but not standard metric
• Angular velocity (?)
• Direction given by right hand rule
• Curl fingers in direction of rotation (??)
• Thumb points in direction of ?

8
Angular measure

• Instantaneous linear velocity (v) is in a
  straight line tangent to the circular path
• v r?
• For constant angular velocity, tangential
  (linear) velocity increases with the distance
  from the origin
• How CDs work

9
Angular measure

• Period (T)time required for 1 complete
  revolution (cycle)
• Units of seconds (s)
• Frequency (f)number of revolutions completed
  every second
• Units of 1/s, s-1, cycles/s
• Renamed Hertz (Hz)

10
Angular Kinematics

• Angular acceleration (?) rate of change of
  angular velocity
• Does not apply to Uniform Circular Motion
• Related to linear (tangential) acceleration

Example 7.10, p. 225
11
Angular Kinematics

• Angular Kinematic Equations

12
Rolling motion

• Rolling without slippingsurface do not move with
  respect to the point of contact
• Direct relationship between rotational and
  translational motion
• s distance traveled by the center of mass
• vcm velocity of the center of mass
• acm acceleration of the center of mass

13
Torque

• Force is the cause of translational acceleration
• Torque(?) is the cause of rotational acceleration
• Units of mN or Nm (not Joules)
• The meter-Newton or Newton-meter
• Moment arm (r)distance from the axis of rotation
  to the the point where force is applied
• Only component of force perpendicular to moment
  arm contributes to rotation

Muscle Torque
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Equilibrium

• Translational equilibrium
• ?F 0
• No net force
• Rotational equilibrium
• ?? 0
• No net torque
• Static Equilibriumno movement
• Kinetic Equilibriumconstant velocity

See-saw Meter stick, page 258
15
Rotational Dynamics

• Newtons 2nd Law (translational)
• ?F mass x acceleration
• Newtons 2nd Law (rotational)
• ?? rotational equivalent of mass x rotational
  equivalent of acceleration

16
Rotational Dynamics

• I moment of inertia
• Rotational equivalent of mass
• Distribution of mass
• Dependent on axis of rotation

common geometrical shapes, p. 304
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Moment of Inertia
18
Moment of Inertia
19
Rotational Kinetic Energy

• Remember, all your translational equations have
  rotational equivalents
• Rotational work
• Rotational KE
• Rotational Work-Energy Theorem