MOMENT OF INERTIA

If m₁, m₂ , ..., m_n are the masses of the particles of a rigid body and r₁, r₂ , ..., r_n are their perpendicular distances from a given axis respectively, then the moment of inertia of the body corresponding to the given axis is given by

                                                            _n

I = m ₁ r ₁ ² + m ₂ r ₂ ² + ... + m _n r _n ² =Σ mi r_i²  

                                                                      ⁱ⁼¹

The magnitude of moment of inertia depends on the selection of the axis and the distribution of mass about it. Its S I unit is kg m2 and dimensional formula is M ¹ L ² T ⁰.

The equations L = Iω and t = I α are analogous to the equations of linear motion P = mv and F = ma respectively which shows that the moment of inertia plays the same role in rotational motion as the mass plays in linear motion. The moment of inertia is the inertia for rotational motion just as the mass is the inertia for linear motion.

TWO THEOREMS REGARDING MOMENT OF INERTIA

( i ) Parallel axes theorem:

The moment of inertia ( I ) of a body about a given axis is equal to the sum of its moment of inertia I c about a parallel axis passing through its center of mass and the product of its mass and square of perpendicular distance ( d ) between the two axes.

I = I _c + M d ²

( ii ) Perpendicular axes theorem:

( a ) For laminar bodies:

For laminar bodies, the moment of inertia I _z about Z-axis normal to its plane is equal to the sum of its moments of inertia about X-axis, I _x and Y-axis, I _y.

I _z = I _x + I _y

( b ) For three-dimensional bodies:

The sum of moments of inertia of a three dimensional body about any three mutually perpendicular axes drawn through the same point is equal to twice the moment of inertia of the body about that point.

I _x + I _y + I _z = 2 I ₀