Torque and Motion Relationships

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Torque and Motion Relationships

• Chapters 9 and 10

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Angular Speed and Velocity

• A body rotating about some axis has angular
  speed.
• ANGULAR SPEED defines how fast the body is
  changing its angular position.
• Angular Speedangular displacement/t

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Angular Speed and Velocity-Continued

• If a direction is specified for the rotating
  body, the body is said to have ANGULAR VELOCITY
  (Vector quantity)

4
Linear Velocity of a Point on a Rotating Body

• The linear distance and speed of some point on a
  rotating segment depends on the distance that
  point is from the axis of rotation (radius of
  rotation).
• The greater the radius, the greater the distance
  the point moves as the segment rotates.

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Continued

• Linear distance (d)
• Radius of rotation r) x angular displacement (q)

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Continued

• Linear speed/velocity
• Radius or rotation r) x angular velocity v)
• THE GREATER the ANGULAR VELOCITY or the GREATER
  the RADIUS OF ROTATION, the GREATER THE ROTATING
  POINTS LINEAR SPEED/VELOCITY. See figure H-1

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Angular Acceleration

• Angular acceleration the rate at which a
  bodys angular direction or speed is changed
  (how fast a body changes speed or direction or
  both)
• Angular acceleration Change in angular
  velocity/time it took to change.
• av2-v1/t

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Continued

• A rigid body, segment, or implement experiences
  an angular acceleration or deceleration only
  during the time a net external torque is applied.
• The acceleration is always in the direction of
  the net torque acting.
• The greater the torque, the greater the angular
  acceleration.

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Relationship of Torque, Rotational Inertia and
Angular Acceleration

• Same principle as linear acceleration- Newtons
  Second Law
• The angular acceleration of a body is directly
  proportional to the net torque applied to it, and
  the bodys acceleration is inversely proportional
  to the rotational inertia of the body.
• aT/I or T I/a

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Continued

• What is Rotational Inertia?
• The measure of resistance to angular acceleration
  of a body.
• The combination of a systems mass and its mass
  distribution determines its rotational inertia.
• I m x r2

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Continued

• The greater a systems mass or the farther from
  the axis of rotation the mass is distributed or
  both, the greater the systems rotational
  inertia.
• The mass distribution factor is more influential
  since its quantity is squared in the equation.
  Figure H-3.

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Radius of Gyration

• Term used when dealing with human bodies (k).
• The distance that represents how far away from
  the axis of rotation a bodys mass is.

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Continued

• A rotating body whose mass is spread away from
  the axis has a large radius of gyration.
• Layout flip in gymnastics/Skater with arms
  outreached.

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Continued

• A body whose mass is packed in close to the axis
  has a small radius of gyration.
• A tucked flip in gymnastics/skater with arms
  pulled in.

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Continued

• Large radius of gyration produces large
  rotational inertia while a small radius of
  gyration produces a small rotational inertia.
• When working with real bodies I mk2

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Continued

• Every time a body segment is rotated, the
  rotational inertia of that segment determines how
  the segment responds to the torque applied to it.
• Our body segments are more massive towards the
  proximal ends, therefore they require less torque
  to angularly accelerate them. See H.4

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aT/I

• Much like linear acceleration.
• If Torque is increased on the same limb, the
  angular acceleration will increase.
• If Rotational Inertia is increased (a limb
  extending from a flexed position), angular
  acceleration will decrease with the same amount
  of Torque applied.
• If Rotational Inertia is increased, the applied
  muscle torque must be increased to produce the
  same angular acceleration.