ENGR 230 Dynamics Sections 17'1

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ENGR 230 Dynamics Sections 17.1
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Moment of Inertia

• The translational aspects of motion are described
  by the equation
• The rotational aspects of motion are described by
  the equation

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Golf Ball Drop Demo
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Moment of Inertia

• Just as mass is a measure of a bodys resistance
  to acceleration inertia is a measure of a bodys
  resistance of a body to angular acceleration.

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What is the purpose of the flywheel on this 1942
John Deere Model G tractor?
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Definition of Moment of Inertia

• We define the moment of inertia as the integral
  of the second moment about an axis of all the
  elements of mass dm which compose a body?
• r moment arm or perpendicular distance to the
  axis of rotation

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Definition of Moment of Inertia
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Differential Volume

• To obtain the moment of inertia we will only
  consider symmetrical bodies having surface which
  can be generated by revolving a curve about an
  axis.

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Shell Elements

• If the shell having a height z, radius r, and a
  thickness dy is chosen then the volume is given
  by
• We need z as a function of y to integrate.

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Disk Elements

• If the disk having a radius r and a thickness dz
  is chosen then the volume is given by
• We need y as a function of z to integrate.

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Solid Cylinder
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Example 17.2
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Parallel Axis Theorem
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Slender Rod

• Find the Moment of Inertia for a slender rod
  length, L about its center of mass.

• Find the Moment of Inertia for a slender rod
  length, L about its end.

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Back to the golf ball drop!
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Radius of Gyration

• Occasionally, the moment of inertia of a body
  about a specified axis is reported as the radius
  of gyration, k.

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Lecture Example The pendulum consists of two
slender rods AB and OC which have a mass of 3
kg/m. The thin plate has a mass of 12 kg/m2.
Determine the center of mass of the pendulum and
the moment of inertia about an axis perpendicular
to the page at Point G.