MOMENT%20OF%20INERTIA

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MOMENT OF INERTIA

• BY
• GP CAPT NC CHATTOPADHYAY

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WHAT IS MOMENT OF INERTIA?

• IT IS THE MOMENT REQUIRED BY A SOLID BODY TO
  OVERCOME ITS RESISTANCE TO ROTATION
• IT IS RESISTANCE OF BENDING MOMENT OF A BEAM
• IT IS THE SECOND MOMENT OF MASS (mr2) OR SECOND
  MOMENT OF AREA (Ar2)
• ITS UNIT IS m4 OR kgm2

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PERPENDICULAR AXIS THEOREM

•   The moment of inertia of a plane area about an
  axis normal to the plane is equal to the sum of
  the moments of inertia about any two mutually
  perpendicular axes lying in the plane and passing
  through the given axis.  Moment of Inertia Iz
  IxIy

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PARALLEL (TRANSFER)AXIS THEOREM

• THE MOMENT OF AREA OF AN OBJECT ABOUT ANY AXIS
  PARALLEL TO THE CENTROIDAL AXIS IS THE SUM OF MI
  ABOUT ITS CENTRODAL AXIS AND THE PRODUCT OF AREA
  WITH THE SQUARE OF DISTANCE OF CG FROM THE REF
  AXIS
• IXX IGAd2
• A is the cross-sectional area. is the
  perpendicuar distance between the centroidal axis
  and the parallel axis.

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Moment of Inertia - Parallel Axis Theorem
Parallel axis theorem Consider the moment of
inertia Ix of an area A with respect to an axis
AA. Denote by y the distance from an element of
area dA to AA.
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Moment of Inertia - Parallel Axis Theorem
Consider an axis BB parallel to AA through the
centroid C of the area, known as the centroidal
axis. The equation of the moment inertia becomes

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Moment of Inertia - Parallel Axis Theorem
The first integral is the moment of inertia about
the centroid.
The second component is the first moment area
about the centroid
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Moment of Inertia - Parallel Axis Theorem
Modify the equation obtained with the parallel
axis theorem.
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Example Moment of Inertia
Compute the moment of inertia in the x about the
AA plane.
AA
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Example Moment of Inertia
Compute the moment of inertia in the x about the
AA plane.
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Example Moment of Inertia
From earlier lecture, the moment of inertia about
the centroid
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Example Moment of Inertia
Using the parallel axis theorem
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Parallel Axis - Why?
Recall that the method of finding centroids of
composite bodies? - Follow a Table technique
How would you be able to find the moment of
inertia of the body. Use a similar technique,
table method, to find the moment of inertia of
the body.
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Parallel Axis - Why?
Use a similar technique, table method, to find
the moment of inertia of the body.
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Moment of Inertia
Use a set of standard tables
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Example - Moment of Inertia
Find the moment of inertia of the body, Ix and
the radius of gyration, kx (rx)
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I THINK. I CAN THINK.
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Example - Moment of Inertia
Set up the reference axis at AB and find the
centroid
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Example - Moment of Inertia
From the table to find the moment of inertia
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Example - Moment of Inertia
Compute the radius of gyration, rx.
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Example - Moment of Inertia
Find the moment of inertia of the body, Ix and
the radius of gyration, kx (rx)
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Example - Moment of Inertia
The components of the two bodies and subtract the
center area from the total area.
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Example - Moment of Inertia
Compute the radius of gyration, rx.
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PROBLEM.

• FIND THE MI OF A CIRCULAR LAMINA OF RADIUS r?