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Perpendicular Axis Theorem




Perpendicular axis theorem


  • This theorem is applicable only to the plane laminar bodies
  • This theorem states that, the moment of inertia of a plane laminar about an axis perpendicular to its plane is equal to the sum of the moment of inertia of the lamina about two axis mutually perpendicular to each other in its plane and intersecting each other at the point where perpendicular axis passes through it
  • Consider plane laminar body of arbitrary shape lying in the x-y plane as shown below in the figure

    Theorems of Moment of Inertia:Perpendicular axis theorem

  • The moment of inertia about the z-axis equals to the sum of the moments of inertia about the x-axis and y axis
  • To prove it consider the moment of inertia about x-axis


    where sum is taken over all the element of the mass mi
  • The moment of inertia about the y axis is

  • Moment of inertia about z axis is

    where ri is perpendicular distance of particle at point P from the OZ axis
  • For each element
    ri2=xi2 + yi2

    Moment of inertia formula for Perpendicular axis theorem


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