Basics of Moment of Inertia

Basics of Moment of Inertia

What is moment of inertia?

Moment of inertia is defined as the quality exhibited by the body to resist the angular acceleration. It is the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation.

What is the formula of the moment of inertia?

Following is the general formula of the moment of inertia that is used in every situation:

Factors affecting the moment of inertia

Following are the factors that affect the moment of inertia:

  • The density of the object
  • Shape and size of the object
  • Configuration of rotation of the object

Moment of inertia units

The moment of inertia units are:

  • Area of moment of inertia: mm4 or in4
  • Mass moment of inertia: kg.m2 or ft.lb.s2
  • Dimensional formula: M1L2T0
  • SI unit: kg.m2

Moment of inertia for different rigid bodies

  • Solid sphere: I = (⅖)MR2
  • Hollow thin-walled sphere: I = (⅔)MR2
  • Solid cylinder: I = (½)MR2
  • Hollow thin-walled cylinder: I = MR2
  • Hollow cylinder:
  • Rectangular plate (axis through the centre): I = (1/12) M (a2 + b2)
  • Rectangular plate (axis along the edge): I = (⅓) Ma2 
  • Slender rod (axis through the centre): I = (1/12) ML2
  • Slender rod (axis through one end): I = (⅓) ML2 

Theorems related to the moment of inertia

There are two theorems associated with the moment of inertia and they are:

  • Perpendicular axis theorem
  • Parallel axis theorem

Perpendicular axis theorem:

Perpendicular axis theorem states that for any planar laminar the moment of inertia of the lamina axis perpendicular to the plane is equal to the sum of moment of inertia of the two axes that lie in the plane which is perpendicular to each other. All three axes must pass through the centre of mass of lamina. The mathematical expression is:

Iz = Ix + Iy 

Where,

  • Iz is the moment of inertia axis perpendicular to the plane of the disc which passes through the centre of mass
  • Ix and Iy are the moments of inertia axes perpendicular in the plane of the lamina which passes through the centre of mass

Parallel axis theorem:

Parallel axis theorem states that for a body of mass m with the moment of inertia at the centre of the mass ICoM and the moment of inertia of the second axis which is parallel at a distance d is given as:

I = ICoM + md2

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