Dynamics of Rotational Motion

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Dynamics of Rotational Motion
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Cross Product
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Cross Product

• The Cross Product (or Vector Product) of two
  vectors A and B is a multiplication of vectors
  where the result is a vector quantity C with a
  direction perpendicular to both vectors A and B,
  and the magnitude equal to ABsin?

Magnitude of the cross (vector) product of two
vectors A and B

• A is the magnitude of the first vector, B is the
  magnitude of the second vector and f is the angle
  between the two vectors.
• The direction of the cross product is
  perpendicular to the plane formed by the two
  vectors in the product. This leaves two possible
  choices which are resolved by using the Right
  Hand Rule.

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Cross Product

• Properties of the Cross Product
• Cross Product is Anti-Commutative.
• Parallel Vectors have Cross Product of zero.
• Cross Product obeys the Distributive Law.
• Product Rule for Derivative of a Cross Product.

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Cross Product

• The above formula for the cross product is useful
  when the magnitudes of the two vectors and the
  angle between them are known.
• If you only know the components of the two
  vectors

Components of cross product vector
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Cross Product

• Right-handed coordinate system, in which

• Left-handed coordinate system, in which

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Torque

• Net force applied to a body gives that body an
  acceleration.
• What does it take to give a body angular
  acceleration?
• Force is required! It must be applied in a way
  that gives a twisting or turning action.

The quantative measure of the tendency of a force
to cause or change the rotational motion of a
body is called torque.

• This body can rotate about axis through O, ? to
  the plane. It is acted by three forces (in the
  plane of figure). The tendency of any force to
  cause the rotation depends on its magnitude and
  on the perpendicular distance (lever arm) between
  the line of action of the force and point O.

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Torque
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Torque

• Torque is a vector quantity that measures the
  tendency of a force to rotate an object about an
  axis. The magnitude of the torque produced by a
  force is defined as

• where r distance between the pivot point and
  the point of application of the force. F the
  magnitude of the force. f the angle between
  the force and a line extending thru the pivot and
  the point of application. Ftan F sin(f) the
  component of the force perpendicular to the line
  connecting the pivot and the point of
  application. L r sin(f) moment arm or lever
  arm distance from the pivot to the line of
  action of the force.

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Torque
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Torque

• Some important points about torque
• Torque has units of Nm. Despite the fact that
  this unit is the same as a Joule it is customary
  to leave torque expressed in Nm (or
  footpounds).
• Engineers will often use the term "moment" to
  describe what physicists call a "torque".
• We will adopt a convention that defines torques
  that tend to cause clockwise rotation as negative
  and torques that tend to cause counter-clockwise
  rotation as positive.
• Torques are always defined relative to a point.
  It is incorrect to simply say the "torque of F".
  Instead you must say the "torque of F relative to
  point X".
• More general definition for the torque is given
  by the vector (or cross product). When a force
  acting at a point which has position vector r
  relative to an origin O the torque exerted by the
  force about the origin is defined as

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Torque
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Torque and Angular Acceleration for a Rigid Body
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Torque and Angular Acceleration for a Rigid Body

• If we consider a rigid body rotating about a
  fixed axis as made up of a collection of
  individual point particles, all of which have to
  obey Newton's Second Law for a particle, then we
  can show that the net torque acting on the body
  about the given axis of rotation will equal the
  moment of inertia of the body about that axis
  times the angular acceleration.

• Z-axis is the axis of rotation the first
  particle has mass m1 and distance r1 from the
  axis of rotation.
• The net force acting on this article has a
  component F1,rad along the radial direction, a
  component F1,tan that is tangent to the circle of
  radius r1 in which particle moves, and component
  F1z along axis of rotation.
• N2L for tangential component is

F1,rad and F1z do not contribute to the torque
about z-axis.
For all particles
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Torque and Angular Acceleration for a Rigid Body

• This expression is the rotational form of
  Newton's Second Law for rigid body motion (for a
  fixed axis of rotation)

N2L for a rigid body in rotational form

• Valid only for rigid bodies! If the body is not
  rigid like a rotating tank of water, the angular
  acceleration is different for different
  particles.
• ?z must be measured in rad/s2 (we used atanr?z
  in derivation)
• The torque on each particle is due to the net
  force on that particle, which is the vector sum
  of external and internal forces.
• According to N3L, the internal forces that any
  pair of particles in the rigid body exert on each
  other are equal and opposite. If these forces act
  along the line joining the two particles, their
  lever arms with respect to any axis are also
  equal. So the torques for each pair are equal and
  opposite and add to ZERO.

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Torque and Angular Acceleration for a Rigid Body
ONLY external torques affect the rigid bodys
rotation!
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N2L in Rotational Form
Rotational Dynamics for Rigid Bodies
Problem-Solving Strategy

• IDENTIFY the relevant concepts
• The equation ??zI?z is useful whenever torques
  act on a rigid body - that is, whenever forces
  act on a rigid body in such a way as to change
  the state of the bodys rotation.
• In some cases you may be able to use an energy
  approach instead. However, if the target variable
  is a force, a torque, an acceleration, an angular
  acceleration, or an elapsed time, the approach
  using equation ??zI?z is almost always the most
  efficient one.

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N2L in Rotational Form
Problem-Solving Strategy

• SET UP the problem using the following steps
• Draw a sketch of the situation and select the
  body or bodies to be analyzed.
• For each body, draw a free-body diagram isolating
  the body and including all the forces (and only
  those forces) that act on the body, including its
  weight. Label unknown quantities with algebraic
  symbols. A new consideration is that you must
  show the shape of the body accurately, including
  all dimensions and angles you will need for
  torque calculations.
• Choose coordinate axes for each body and indicate
  a positive sense of rotation for each rotating
  body. If there is a linear acceleration, its
  usually simplest to pick a positive axis in its
  direction. If you know the sense of ?z in
  advance, picking it as the positive sense of
  rotation simplifies the calculations. When you
  represent a force in terms of its components,
  cross out the original force to avoid including
  it twice.

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N2L in Rotational Form
Problem-Solving Strategy

• EXECUTE the solution as follows
• For each body in the problem, decide whether it
  undergoes translational motion, rotational
  motion, or both. Depending on the behavior of the
  body in question, apply ?Fma, ??zI?z, or both
  to the body. Be careful to write separate
  equations of motion for each body.
• There may be geometrical relations between the
  motions of two or more bodies, as with a string
  that unwinds from a pulley while turning it or a
  wheel that rolls without slipping. Express these
  relations in algebraic form, usually as relations
  between two linear accelerations or between a
  linear acceleration and an angular acceleration.
• Check that the number of equations matches the
  number of unknown quantities. Then solve the
  equations to find the target variable(s).

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N2L in Rotational Form
Problem-Solving Strategy

• EVALUATE your answer
• Check that the algebraic signs of your results
  make sense.
• As an example, suppose the problem is about a
  spool of thread. If you are pulling thread off
  the spool, your answers should not tell you that
  the spool is turning in the direction that rolls
  the thread back on the spool!
• Whenever possible, check the results for special
  cases or extreme values of quantities and compare
  them with your intuitive expectations.
• Ask yourself Does this result make sense?

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Rigid-Body Rotation about a Moving Axis
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Rigid-Body Rotation about a Moving Axis

• Lets extend analysis of rotational motion to
  cases in which the axis of rotation moves the
  motion of a body is combined translation and
  rotation.
• Every possible motion of a rigid body can be
  represented as a combination of translational
  motion of the center of mass and rotation about
  an axis through the center of mass.
• It is applicable even when the center of mass
  accelerates (so that is not at rest in any
  inertial frame).
• The translation of the center of mass and the
  rotation about the center of mass can be treated
  as separate but related problems.
• The prove of all that is beyond of the scope of
  this course. We will learn concept only.

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Rigid-Body Rotation about a Moving Axis

• If a round object of cross-sectional radius R
  rolls without slipping then the distance along
  the surface that the object covers will be the
  same as the arc length along the edge of the
  circular object that has been in contact with the
  surface (i.e. s Rq).
• Differentiating this expression with respect to
  time shows that the speed of the center of mass
  of the object will be given by

Condition for rolling without slipping

• This condition must be satisfied if an object is
  rolling without slipping.
• Rolling motion can be thought of in two different
  ways
• Pure rotation about the instantaneous point of
  contact (P) of the object with the surface.
• Superposition of translation of the center of
  mass plus rotation about the center of mass.

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Rigid-Body Rotation about a Moving Axis

• The wheel is symmetrical, so its CM is at its
  geometric center.
• We view the motion in inertial frame of reference
  in which the surface is at rest. In order not to
  slip, the point of contact (where the wheel
  contacts the ground) is instantaneously at rest
  as well. Hence the velocity of the point of
  contact relative to the CM must have the same
  magnitude but opposite direction as the CM
  velocity.
• If the radius of the wheel is R and its angular
  speed about CM is ? vcmR?.

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Rigid-Body Rotation about a Moving Axis

• The velocity of a point on the wheel is the
  vector sum of the velocity of CM and the velocity
  of the point relative to the center of mass.
• Thus, the point of the contact is instantaneously
  at rest, point 3 at the top of the wheel is
  moving forward twice as fast as the center of
  mass points 2 and 4 at the sides have velocities
  at 45 degrees to the horizontal.

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Rigid-Body Rotation about a Moving Axis

• The kinetic energy of an object that is rolling
  without slipping is given by the sum of the
  rotational kinetic energy about the center of
  mass plus the translational kinetic energy of the
  center of mass

Rigid body with both translation and rotation

• If a rigid body changes height as it moves, you
  must also consider gravitational potential energy
  
• The gravitational potential energy associated
  with any extended body of mass M, rigid or not,
  is the same as if you replace the body by a
  particle of mass M located at the bodys center
  of mass

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Rigid-Body Rotation about a Moving Axis

• Dynamics of Combined Translation and Rotation
• The combined translational and rotational motion
  of an object can also be analyzed from the
  standpoint of dynamics. In this case the object
  must obey both of the following forms of Newton's
  Second Law

• Two following conditions should be met
• The axis through the center of mass must be an
  axis of symmetry
• The axis must not change direction

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Rolling Friction
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Work and Power in Rotational Motion

• The work done by a torque on an object that
  undergoes an angular displacement from q1 to q2
  is given by

Work done by a torque

• If the torque is constant then the work done is
  given by

Work done by a constant torque

• Note similarity between these expressions and
  the equations for work done by a force (WFS).

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Work and Power in Rotational Motion

• The rotational analog to the Work - Energy
  Theorem is

• The change in rotational kinetic energy of a
  rigid body equals the work done by forces exerted
  from outside the body.
• The rate at which work is performed is the power

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Angular Momentum
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Angular Momentum of a Particle. Definition

• The angular momentum L of a particle relative to
  a point O is the cross product of the particle's
  position r relative to O with the linear momentum
  p of the particle.

Mass m is moving in XY plane
Angular momentum of a particle

• The value of the angular momentum depends on the
  choice of the origin O, since it involves the
  position vector relative to the origin
• The units of angular momentum kgm2/s

Lever arm
Right-hand rule
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Angular Momentum of a Particle

• When a net force F acts on a particle, its
  velocity and linear momentum change. Thus,
  angular momentum may also change.

Vector product of vector by itself 0
For a particle acted on by net force F

• Rate of change of angular momentum L of a
  particle equals the torque of the net force
  acting on it.

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Angular Momentum of a Rigid Body

• Rigid body rotating about Z-axis with angular
  speed ?
• Consider a thin slice of the body lying in XY
  plane

• Each particle in the slice moves in a circle
  centered in the origin O, and its velocity vi at
  each instant ? to its position vector ri
• Thus, ?90, and particle of the mass mi at
  distance ri from O has speed viri?
• The direction of angular momentum Li is by
  right-hand rule and the magnitude

• The total angular momentum of the slice is the
  sum of Li of particles

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Angular Momentum of a Rigid Body

• For points not lying in XY plane, the position
  vectors have components in Z-direction as well as
  in X- and Y-directions. This gives the angular
  momentum of each particle a component
  perpendicular to Z-axis. But if Z-axis is the
  axis of symmetry, ?- components for particles on
  opposite sides of this axis add up to ZERO.

• Thus, when a rigid body rotates about an axis of
  symmetry, its angular momentum vector L lies
  along the symmetry axis, and its magnitude is
  LI?
• The angular velocity vector lies also along the
  rotation axis. Hence for a rigid body rotating
  around axis of symmetry, L and ? have the same
  direction

for a rigid body rotating around a symmetry axis
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Angular Momentum of a Rigid Body
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Angular Momentum and Torque

• For any system of particles (including both rigid
  and non-rigid bodies), the rate of change of the
  total angular momentum equals the sum of the
  torques of all forces acting on all the particles
  
• Torques of the internal forces add to zero if
  these forces act along the line from one particle
  to another, so the sum of torques includes only
  the torques of external forces

for any system of particles

• If the system of particles is a rigid body
  rotating about its axis of symmetry (Z-axis),
  then LzI?z and I is constant. If this axis is
  fixed in space, then the vectors L and ? change
  only in magnitude, not in direction

• If body is not rigid, I may change, and L changes
  even if ? is constant

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Angular Momentum of a Rigid Body

• If the axis of rotation is not a symmetry axis, L
  does not in general lie along the rotation axis.
• Even if ? is constant, the direction of L changes
  and a net torque is required to maintain rotation
  
• If the body is an unbalanced wheel of your car,
  this torque is provided by friction in the
  bearings, which causes the bearings to wear out
• Balancing a wheel means distributing the mass so
  that the rotation axis is an axis of symmetry,
  then L points along the rotation axis, and no net
  torque is required to keep the wheel turning

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Conservation of Angular Momentum
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Conservation of Angular Momentum

• When the net external force torque acting on a
  system is zero, the total angular momentum of the
  system is constant (or conserved)

Angular momentum conservation

• This principle is universal conservation law,
  valid at all scales from atomic and nuclear
  systems to the motions of galaxies
• Circus acrobat, diver, ice skater use this
  principle
• Suppose acrobat has just left a swing with arms
  and legs extended and rotating counterclockwise
  about her center of mass. When she pulls her arms
  and legs in, her moment of inertia Icm with
  respect to her center of mass changes from a
  large value I1 to much smaller value I2. The only
  external force is her weight, which has no torque
  (goes through center of mass). So angular
  momentum remains constant, and angular speed
  changes

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Physics of Falling Cats

• How does a cat land on its legs when dropped?
• Moment of inertia is important ...
• To understand how a cat can land on it's feet,
  you must first know some concepts of rotational
  motion, since the cat rotates as it falls.
• Reminder The moment of inertia of an object is
  determined by the distance it's mass is
  distributed from the rotational axis.
• Relating this to the cat, if the cat stretches
  out it's legs and tail, it increases it's moment
  of inertia conversely, it can decrease it's
  moment of inertia by curling up.
• Remember how it was proved by extending your
  professors arms while spinning around on a
  swivel chair?
• Just as a more massive object requires more force
  to move, an object with a greater moment of
  inertia requires more torque to spin. Therefore
  by manipulating it's moment of inertia, by
  extending and retracting its legs and rotating
  its tail, the cat can change the speed at which
  it rotates, giving it control over which part of
  it's body comes in contact with the ground.

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Physics of Falling Cats

• ... and the conservation of angular
  momentum ...
• If a cat is dropped they almost always tend to
  land on their feet because they use the
  conservation of angular momentum to change their
  orientation
• When a cat falls, as you would expect, its centre
  of mass follows a parabolic path. The cat falls
  with a definite angular momentum about an axis
  through the cats centre of mass.
• When the cat is in the air, no net external
  torque acts on it about its centre of mass, so
  the angular momentum about the cats centre of
  mass cannot change.
• By pulling in its legs, cat can considerably
  reduce it rotational inertia about the same axis
  and thus considerably increase its angular speed.
  
• Stretching out its legs increases its rotational
  inertia and thus slows the cats angular speed.
• Conservation of angular momentum allows cat to
  rotate its body and slow its rate of rotation
  enough so that it lands on its feet

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Conservation of Angular Momentum

• Falling cat twists different parts of its body in
  different directions so that it lands feet first
• At all times during this process the angular
  momentum of the cat as a whole is zero
• A free-falling cat cannot alter its total angular
  momentum. Nonetheless, by swinging its tail and
  twisting its body to alter its moment of inertia,
  the cat can manage to alter its orientation

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Falling Cats More Information

• How does a cat land on its legs when dropped?
• Cats have the seemingly unique ability to orient
  themselves in a fall allowing them to avoid many
  injuries. This ability is attributed to two
  significant feline characteristics righting
  reflex and unique skeletal structure.
• The righting reflex is the cats ability to
  first, know up from down, and then the innate
  nature to rotate in mid air to orient the body so
  its feet face downward.
• Animal experts say that this instinct is
  observable in kittens as young as three to four
  weeks, and is fully developed by the age of seven
  weeks.
• A cats righting reflex is augmented by an
  unusually flexible backbone and the absence of a
  collarbone in the skeleton. Combined, these
  factors allow for amazing flexibility and upper
  body rotation. By turning the head and forefeet,
  the rest of the body naturally follows and cat is
  able reorient itself.
• Like many small animals, cats are said to have a
  non-fatal terminal falling velocity. That is,
  because of their very low body volume-to-weight
  ratio these animals are able to slow their decent
  by spreading out (flying squirrel style). Animals
  with these characteristics are fluffy and have a
  high drag coefficient giving them a greater
  chance of surviving these falls.

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How does a Speedometer work?

• Analog speedometer translates the high-speed
  rotation of a permanent magnet into the slow,
  damped motion of a spring-loaded shaft.
• A needle on this shaft indicates speed in mph or
  km/h. The magnet turns within a movable drag cup
  made of a nonmagnetic metal. As the magnet
  rotates, it exerts a magnetic force on the
  movable cup that tends to turn it against the
  restraint of a spiral spring. As the magnet
  rotates faster, the pull on the cup increases so
  needle indicates a higher speed.
• You will study magnetic forces later !