Rotation and Torque

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Rotation and Torque

• Lecture 09
• Thursday 12 February 2004

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ROTATION DEFINITIONS

• Angular position q
• Angular displacement q 2 q 1 Dq

• Instantaneous Angular velocity

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What is the direction of the angular velocity?

• Use your right hand
• Curl your fingers in the direction of the
  rotation
• Out-stretched thumb points in the direction of
  the angular velocity

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DEFINITIONS (CONTINUED)
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Direction of Angular Acceleration

• The easiest way to get the direction of the
  angular acceleration is to determine the
  direction of the angular velocity and then
• If the object is speeding up, velocity and
  acceleration must be in the same direction.
• If the object is slowing down, velocity and
  acceleration must be in opposite directions.

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For constant a

• x q
• v w
• a a

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Relating Linear and Angular Variables
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Three Accelerations

• Centripetal Acceleration
• (radial component of the linear acceleration)
• -always non-zero in circular motion.
• Tangential Acceleration
• (component of linear acc. along the direction of
  the velocity)
• -non-zero if the object is speeding up or slowing
  down.
• Angular Acceleration
• (rate of change in angular velocity)
• -non-zero is the object is speeding up or slowing
  down.

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Energy Considerations
Although its linear velocity v is zero, the
rapidly rotating blade of a table saw certainly
has kinetic energy due to that rotation. How
can we express the energy? We need to treat the
table saw (and any other rotating rigid body) as
a collection of particles with different linear
speeds.
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KINETIC ENERGY OF ROTATION
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Defining Rotational Inertia

• The larger the mass, the smaller the acceleration
  produced by a given force.
• The rotational inertia I plays the equivalent
  role in rotational motion as mass m in
  translational motion.

• I is a measure of how hard it is to get an object
  rotating. The larger I, the smaller the angular
  acceleration produced by a given force.

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Determining the Rotational Inertia of an Object
I is a function of both the mass and shape of the
object. It also depends on the axis of rotation.

• For common shapes, rotational inertias are listed
  in tables. A simple version of which is in
  chapter 11 of your text book.
• For collections of point masses, we can use
• where r is the distance from the axis (or point)
  of rotation.
• For more complicated objects made up of objects
  from 1 or 2 above, we can use the fact that
  rotational inertia is a scalar and so just adds
  as mass would.

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Comparison to Translation

• x q
• v w
• a a
• m I
• K1/2mv2?1/2I?2

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Force and Torque
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Torque as a Cross Product

• (Like FMa)
• The direction of the Torque is always in the
  direction of the angular acceleration.
• For objects in equilibrium, ??0 AND ?F0

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Torque Corresponds to Force

• Just as Force produces translational acceleration
  (causes linear motion in an object starting at
  rest, for example)
• Torque produces rotational acceleration (cause a
  rotational motion in an object starting from
  rest, for example)
• The cross or vector product is another way to
  multiply vectors. Cross product results in a
  vector (e.g. Torque). Dot product (goes with
  cos ?) results in a scalar (e.g. Work)
• r is the vector that starts at the point (or
  axis) of rotation and ends on the point at which
  the force is applied.

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An Example
Forces on extended bodies can be viewed as
acting on a point mass (with the same total
mass) At the objects center of mass (balancing
point)
x
W
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Determining Direction of A CROSS PRODUCT
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Angular Momentum of a Particle

• Angular momentum of a particle about a point of
  rotation
• This is similar to Torques

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Find the direction of the angular momentum
vector-Right hand rule
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Does an object have to be moving in a circle to
have angular momentum?

• No.
• Once we define a point (or axis) of rotation
  (that is, a center), any object with a linear
  momentum that does not move directly
  through that point has an angular momentum
  defined relative to the chosen center as

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