Rotational Motion

1
Chapter 10

• Rotational Motion

2
Rigid Object

• A rigid object is one that is nondeformable
• The relative locations of all particles making up
  the object remain constant
• All real objects are deformable to some extent,
  but the rigid object model is very useful in many
  situations where the deformation is negligible

3
Angular Position

• Axis of rotation is the center of the disc
• Choose a fixed reference line
• Point P is at a fixed distance r from the origin

4
Angular Position, 2

• Point P will rotate about the origin in a circle
  of radius r
• Every particle on the disc undergoes circular
  motion about the origin, O
• Polar coordinates are convenient to use to
  represent the position of P (or any other point)
• P is located at (r, q) where r is the distance
  from the origin to P and q is the measured
  counterclockwise from the reference line

5
Angular Position, 3

• As the particle moves, the only coordinate that
  changes is q
• As the particle moves through q, it moves though
  an arc length s.
• The arc length and r are related
• s q r

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Radian

• This can also be expressed as
• q is a pure number, but commonly is given the
  artificial unit, radian
• One radian is the angle subtended by an arc
  length equal to the radius of the arc

7
Conversions

• Comparing degrees and radians
• Converting from degrees to radians

8
Angular Position, final

• We can associate the angle q with the entire
  rigid object as well as with an individual
  particle
• Remember every particle on the object rotates
  through the same angle
• The angular position of the rigid object is the
  angle q between the reference line on the object
  and the fixed reference line in space
• The fixed reference line in space is often the
  x-axis

9
Angular Displacement

• The angular displacement is defined as the angle
  the object rotates through during some time
  interval
• This is the angle that the reference line of
  length r sweeps out

10
Average Angular Speed

• The average angular speed, ?, of a rotating rigid
  object is the ratio of the angular displacement
  to the time interval

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Instantaneous Angular Speed

• The instantaneous angular speed is defined as the
  limit of the average speed as the time interval
  approaches zero

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Angular Speed, final

• Units of angular speed are radians/sec
• rad/s or s-1 since radians have no dimensions
• Angular speed will be positive if ? is increasing
  (counterclockwise)
• Angular speed will be negative if ? is decreasing
  (clockwise)

13
Average Angular Acceleration

• The average angular acceleration, a,
  
• of an object is defined as the ratio of the
  change in the angular speed to the time it takes
  for the object to undergo the change

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Instantaneous Angular Acceleration

• The instantaneous angular acceleration is defined
  as the limit of the average angular acceleration
  as the time goes to 0
• Units of angular acceleration are rad/s2 or s-2
  since radians have no dimensions

15
Angular Motion, General Notes

• When a rigid object rotates about a fixed axis in
  a given time interval, every portion on the
  object rotates through the same angle in a given
  time interval and has the same angular speed and
  the same angular acceleration
• So q, w, a all characterize the motion of the
  entire rigid object as well as the individual
  particles in the object

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Directions, details

• Strictly speaking, the speed and acceleration (w,
  a) are the magnitudes of the velocity and
  acceleration vectors
• The directions are actually given by the
  right-hand rule

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Directions, final

• The direction of is along the axis of
  rotation
• By convention, its direction is out of the plane
  of the diagram when the rotation is
  counterclockwise
• its direction is into of the plane of the diagram
  when the rotation is clockwise
• The direction of is the same as if the
  angular speed is increasing and antiparallel if
  the speed is decreasing

18
Rotational Kinematics

• Under constant angular acceleration, we can
  describe the motion of the rigid object using a
  set of kinematic equations
• These are similar to the kinematic equations for
  linear motion
• The rotational equations have the same
  mathematical form as the linear equations

19
Rotational Kinematic Equations
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Comparison Between Rotational and Linear Equations
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Relationship Between Angular and Linear Quantities

• Displacements
• Speeds
• Accelerations

• Every point on the rotating object has the same
  angular motion
• Every point on the rotating object does not have
  the same linear motion

22
Speed Comparison

• The linear velocity is always tangent to the
  circular path
• called the tangential velocity
• The magnitude is defined by the tangential speed

23
Acceleration Comparison

• The tangential acceleration is the derivative of
  the tangential velocity

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Speed and Acceleration Note

• All points on the rigid object will have the same
  angular speed, but not the same tangential speed
• All points on the rigid object will have the same
  angular acceleration, but not the same tangential
  acceleration
• The tangential quantities depend on r, and r is
  not the same for all points on the object

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Centripetal Acceleration

• An object traveling in a circle, even though it
  moves with a constant speed, will have an
  acceleration
• Therefore, each point on a rotating rigid object
  will experience a centripetal acceleration

26
Resultant Acceleration

• The tangential component of the acceleration is
  due to changing speed
• The centripetal component of the acceleration is
  due to changing direction
• Total acceleration can be found from these
  components

27
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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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)

30
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

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Moments of Inertia of Various Rigid Objects
34
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 the force makes with the
  horizontal
• d is the moment arm (or lever arm)

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Torque, cont

• 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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Torque, final

• 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

37
Net Torque

• 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

38
Torque vs. Force

• Forces can cause a change in linear motion
• Described by Newtons Second Law
• Forces can cause a change in rotational motion
• The effectiveness of this change depends on the
  force and the moment arm
• The change in rotational motion depends on the
  torque

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Torque Units

• The SI units of torque are 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

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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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Problem Solving Strategy Rigid Object in
Equilibrium

• Conceptualize
• Identify the forces acting on the object
• Think about the effect of each force on the
  rotation of the object, if the force was acting
  by itself
• Categorize
• Confirm the object is a rigid object in
  equilibrium

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Problem Solving Strategy Rigid Object in
Equilibrium, 2

• Analyze
• Draw a free body diagram
• Label all external forces acting on the object
• Resolve all the forces into rectangular
  components
• Apply the first condition of equilibrium

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Problem Solving Strategy Rigid Object in
Equilibrium, 3

• Analyze, cont
• Choose a convenient axis for calculating torques
• Choose an axis that simplifies your calculations
• Apply the second condition of equilibrium
• Solve the simultaneous equations
• Finalize
• Be sure your results are consistent with the free
  body diagram
• Check calculations

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Torque and Angular Acceleration on a Particle

• 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

48
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

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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

50
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

51
Angular Momentum

• The instantaneous angular momentum of a
  particle relative to the origin O is defined as
  the cross product of the particles instantaneous
  position vector and its instantaneous linear
  momentum

52
Torque and Angular Momentum

• The torque is related to the angular momentum
• Similar to the way force is related to linear
  momentum
• This is the rotational analog of Newtons Second
  Law
• The torque and angular momentum must be measured
  about the same origin
• This is valid for any origin fixed in an inertial
  frame

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More About Angular Momentum

• The SI units of angular momentum are (kg.m2)/ s
• The axes used to define the torque and the
  angular momentum must be the same
• When several forces are acting on the object, the
  net torque must be used

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Angular Momentum of a System of Particles

• The total angular momentum of a system of
  particles is defined as the vector sum of the
  angular momenta of the individual particles
• Differentiating with respect to time

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Angular Momentum of a Rotating Rigid Object, cont

• To find the angular momentum of the entire
  object, add the angular momenta of all the
  individual particles
• This is analogous to the translational momentum
  of p m v

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Summary of Useful Equations
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Conservation of Angular Momentum

• The total angular momentum of a system is
  conserved if the resultant external torque acting
  on the system is zero
• Net torque 0 -gt means that the system is
  isolated
• For a system of particles, Ltot SLn constant

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Conservation of Angular Momentum, cont

• If the mass of an isolated system undergoes
  redistribution, the moment of inertia changes
• The conservation of angular momentum requires a
  compensating change in the angular velocity
• Ii wi If wf
• This holds for rotation about a fixed axis and
  for rotation about an axis through the center of
  mass of a moving system
• The net torque must be zero in any case

59
Conservation Law Summary

• For an isolated system -
• (1) Conservation of Energy
• Ei Ef
• (2) Conservation of Linear Momentum
• (3) Conservation of Angular Momentum

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Gyroscope

• Angular momentum is the basis of the operation of
  a gyroscope
• A gyroscope is a spinning object used to control
  or maintain the orientation in space of the
  object or a system containing the object
• Gyroscopes undergo precessional motion
• The symmetry axis rotates, sweeping out a cone

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Precessional Motion of a Gyroscope

• The external forces acting on the top are the
  normal and the gravitational forces
• The torque due to the gravitational force is in
  the xy plane
• Only the direction of the angular momentum
  changes, causing the precession

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Rolling Object

• The red curve shows the path moved by a point on
  the rim of the object
• This path is called a cycloid
• The green line shows the path of the center of
  mass of the object

63
Pure Rolling Motion

• The surfaces must exert friction forces on each
  other
• Otherwise the object would slide rather than roll
• In pure rolling motion, an object rolls without
  slipping
• In such a case, there is a simple relationship
  between its rotational and translational motions

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Rolling Object, Center of Mass

• The velocity of the center of mass is
• The acceleration of the center of mass is

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Rolling Object, Other Points

• A point on the rim, P, rotates to various
  positions such as Q and P
• At any instant, the point on the rim located at
  point P is at rest relative to the surface since
  no slipping occurs

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Total Kinetic Energy of a Rolling Object

• The total kinetic energy of a rolling object is
  the sum of the translational energy of its center
  of mass and the rotational kinetic energy about
  its center of mass
• K 1/2 ICM w2 1/2 MvCM2

67
Parallel-Axis Theorem

• For an arbitrary axis, the parallel-axis theorem
  often simplifies calculations
• The theorem states Ip ICM MD 2
• Ip is about any axis parallel to the axis through
  the center of mass of the object
• ICM is about the axis through the center of mass
• D is the distance from the center of mass axis to
  the arbitrary axis

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Total Kinetic Energy, Example

• Accelerated rolling motion is possible only if
  friction is present between the sphere and the
  incline
• The friction produces the net torque required for
  rotation

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Total Kinetic Energy, Example cont

• Despite the friction, no loss of mechanical
  energy occurs because the contact point is at
  rest relative to the surface at any instant
• Let U 0 at the bottom of the plane
• Kf U f Ki Ui
• Kf 1/2 (ICM / R 2) vCM2 1/2 MvCM2
• Ui Mgh
• Uf Ki 0

70
Turning a Spacecraft

• Here the gyroscope is not rotating
• The angular momentum of the spacecraft about its
  center of mass is zero

71
Turning a Spacecraft, cont

• Now assume the gyroscope is set into motion
• The angular momentum must remain zero
• The spacecraft will turn in the direction
  opposite to that of the gyroscope