Two-Dimensional Rotational Dynamics 8.01 W09D2

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Two-Dimensional Rotational Dynamics8.01W09D2
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Todays Reading Assignment W09D2

• Young and Freedman 1.10 (Vector Product),
  10.1-10.2, 10.4, 11.1-11.3

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Rigid Bodies

• Rigid body An extended object in which the
  distance between any two points in the object is
  constant in time. Examples sphere, disk
• Effect of external forces (the solid arrows
  represent forces)

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Main Idea Fixed Axis Rotation of Rigid Body

• Torque produces angular acceleration about center
  of mass
• is the moment of inertial about the center
  of mass
• is the angular acceleration about center of
  mass

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Recall Fixed Axis Rotation Kinematics

• Angle variable
• Angular velocity
• Angular acceleration
• Mass element
• Radius of orbit
• Moment of inertia
• Parallel Axis Theorem

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Torque as a Vector

• Force exerted at a point P on a rigid
  body.
• Vector from a point S to the point P.

Torque about point S due to the force exerted at
point P
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Summary Cross Product

• Magnitude equal to the area of the parallelogram
  defined by the two vectors
• Direction determined by
• the Right-Hand-Rule

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Torque Magnitude and Direction

• Magnitude of torque about a point S due to force
  acting at point P

Direction of torque Perpendicular to the
plane formed by and .Determined
by the Right-Hand-rule.
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Properties of Cross Products
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Cross Product of Unit Vectors

• Unit vectors in Cartesian coordinates

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Components of Cross Product
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Concept Question Torque

• Consider two vectors with x gt 0 and
• with Fx gt 0 and Fz gt 0 . The cross product
• points in the
• x-direction
• -x-direction
• y-direction
• -y-direction
• z-direction
• -z-direction
• None of the above directions

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Concept Question

• In the figure, a force of magnitude F is applied
  to one end of a lever of length L. What is the
  magnitude of the torque about the point S?
• FL sinq
• FL cosq
• FL tanq
• None of the above

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Recall Rotational Kinematics

• Individual element of mass
• Radius of orbit
• Tangential velocity
• Tangential acceleration
• Radial Acceleration

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Dynamics Newtons Second Law and Torque about S

• Tangential force on mass element produces torque
• Newtons Second Law
• Torque about S
• z-component of torque about S

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Moment of Inertia and Torque

• Component of the total torque about an axis
  passing through S is the sum over all elements
• Recall Moment of Inertia about and axis passing
  through S
• Summary

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Torque due to Uniform Gravitational Force

• The total torque on a rigid body due to the
  gravitational force can be determined by placing
  all the gravitational force at the center-of-mass
  of the object.

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Torque and Static Equilibrium
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Conditions for Static Equilibrium

• (1) Translational equilibrium the sum of the
  forces acting on the rigid body is zero.
• (2) Rotational Equilibrium the vector sum of the
  torques about any point S in a rigid body is zero.

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Concept Question
A 1 kg rock is suspended by a massless string
from one end of a 1 m measuring stick. What is
the mass of the measuring stick if it is balanced
by a support force at the 0.25 m from the left
end?

• 0.25 kg.
• 0.5 kg.
• 1.0 kg.
• 2.0 kg.
• 4.0 kg.
• 6. Impossible to determine.

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Concept Question Tipping

• A box, with its center-of-mass off-center as
  indicated by the dot, is placed on an inclined
  plane. In which of the four orientations shown,
  if any, does the box tip over?

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Problem Solving Strategy Static Equilibrium

• Force
• Identify System and draw all forces and where
  they act on Free Body Force Diagram
• Write down equations for static equilibrium of
  the forces sum of forces is zero
• Torque
• Choose point to analyze the torque about.
• Choose sign convention for torque
• Calculate torque about that point for each force.
  (Note sign of torque.)
• Write down equation corresponding to condition
  for static equilibrium sum of torques is zero

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Table Problem Standing on a Hill
A person is standing on a hill that is sloped at
an angle a with respect to the horizontal. The
persons legs are separated by a distance d, with
one foot uphill and one downhill. The center of
mass of the person is at a distance h above the
ground, perpendicular to the hillside, midway
between the persons feet. Assume that the
coefficient of static friction between the
persons feet and the hill is sufficiently large
that the person will not slip. a) What is the
magnitude of the normal force on each foot? b)
How far must the feet be apart so that the normal
force on the upper foot is just zero? (This is
the instant when the person starts to rotate and
fall over.)
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Fixed Axis Rotational Dynamics
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Worked Example Moment of Inertia Wheel

• An object of mass m is attached to a string
  which is wound around a disc of radius Rd. The
  object is released and takes a time t to fall a
  distance s. What is the moment of inertia of the
  disc?

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Analysis Measuring Moment of Inertia

• Free body force diagrams and force equations
• Rotational equation
• Constraint
• Solve for moment of inertia
• Time to travel distance s

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Demo Moment of Inertia Wheel

• Measuring the moment of inertia.

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Properties of Wheel

• Radius of disc
• Mass of disc
• Mass of weight holder
• Theoretical result

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Rotational Work

• Tangential force
• Displacement vector
• work for a small displacement

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Rotational Work

• Newtons Second Law
• Tangential acceleration
• Work for small displacement
• Summation becomes integration for continuous body

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Rotational Work

• Rotational work for a small displacement
• Torque about S
• Infinitesimal rotational work
• Integrate total work

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Rotational Work-Kinetic Energy Theorem

• Infinitesimal rotational work
• Integrate rotational work
• Kinetic energy of rotation about S

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Rotational Power

• Rotational power is the time rate of doing
  rotational work
• Product of the applied torque with the angular
  velocity

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Concept Question Chrome Inertial Wheel

• A fixed torque is applied to the shaft of the
  chrome inertial wheel. If the four weights on the
  arms are slid out, the component of the angular
  acceleration along the shaft direction will
• increase.
• decrease.
• remain the same.

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Problem Solving StrategyTwo Dimensional Rotation

• Step 1 Draw free body force diagrams for each
  object and indicate the point of application of
  each force
• Step 2 Select point to compute torque about
  (generally select center of mass)
• Step 3 Choose coordinate system. Indicate
  positive direction for increasing rotational
  angle.
• Step 4 Apply Newtons Second Law and Torque Law
  to obtain equations
• Step 5 Look for constraint condition between
  rotational acceleration and any linear
  accelerations.
• Step 6 Design algebraic strategy to find
  quantities of interest

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Rotor Moment of Inertia
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Table Problem Moment of Inertia Wheel

• A steel washer is mounted on a cylindrical
  rotor . The inner radius of the washer is R. A
  massless string, with an object of mass m
  attached to the other end, is wrapped around the
  side of the rotor and passes over a massless
  pulley. Assume that there is a constant
  frictional torque about the axis of the rotor.
  The object is released and falls. As the mass
  falls, the rotor undergoes an angular
  acceleration of magnitude a1. After the string
  detaches from the rotor, the rotor coasts to a
  stop with an angular acceleration of magnitude
  a2. Let g denote the gravitational constant.
• What is the moment of inertia of the rotor
  assembly (including the washer) about the
  rotation axis?

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Solution Moment of Inertia of Rotor

• Force and rotational equations while weight is
  descending
• Constraint
• Rotational equation while slowing down
• Solve for moment of inertia

Speeding up
Slowing down
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Worked Example Change in Rotational Energy and
Work

• While the rotor is slowing down, use work-energy
  techniques to find frictional torque on the
  rotor.

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Next Reading Assignment W09D3

• Young and Freedman 1.10 (Vector Product)
  10.1-10.2, 10.5-10.6 11.1-11.3