Center of Mass Concept

1
Center of Mass Concept

• The center of mass is also referred to as the
  center of gravity. We can locate the center of
  mass (CM) in 2 different ways shown in the next 2
  slides

2
Locating Center of Gravity 1

• The center of gravity can be thought of as a
  point within a body where all of the weight of
  the body is located. Thus we can locate this
  point by balancing the object on a fulcrum, such
  as on the seesaw shown here

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Locating Center of Gravity 2

• The center of gravity will always move to a
  stable equilibrium point directly below a point
  of suspension. Thus we can use the technique
  illustrated in this diagram to experimentally
  locate the CM point

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Calculating the CM point
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Calculating the CM point 2
See Active Figure 9.17 at pse6.com
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Calculating CM point in continuous objects
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Motion of the CM point
The CM point follows the normal parabolic
trajectory of any point object. However, the
other points within the object simultaneously rota
te around the CM point. See Active Figure 9.16
at pse6.com
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Moment of Inertia of Point Objects
Unit kg-m2
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Example of Moment of Inertia for Point Objects
I 2 (mb2 Ma2)
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Calculation of Moment of Inertia for Continuous
Object

• dm ldx for straight line
• dm lRdq for thin ring (hoop)
• l units kg/m

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Moment of Inertia for straight line object
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Moment of Inertia for straight rod about an axis
through CM
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Moment of Inertia for straight rod about an axis
through one end
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Moment of Inertia of Thin Ring (Hoop)
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(No Transcript)
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Moment of Inertia of Cylinder/Disk
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Moment of Inertia of Solid Sphere
See Active Figure 10.30 at pse6.com
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Derivation of Parallel Axis Thm.
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Example of Parallel Axis Thm.

• Given ICM for straight rod (1/12)ML2,
• use the Parallel Axis Thm. to calculate
• the moment of inertia for an axis through
• one end of the rod
• Solution
• I ICM MD2 (1/12)ML2 M(L/2)2
• I (1/12 ¼)ML2 (1/3)ML2
• Note that above result equals the result we get
  when directly integrating.