MOMENT OF INERTIA

Like the centre of mass, the moment of inertia is a property of an object that is related to its mass distribution. The moment of inertia (denoted by I) is an important quantity in the study of system of particles that are rotating. The role of the moment of inertia in the study of rotational motion is analogous to that of mass in the study of linear motion. Moment of inertia gives a measurement of the resistance of a body to a change in its rotational motion. If a body is at rest, the larger the moment of inertia of a body, the more difficult it is to put that body into rotational motion. Similarly, the larger the moment of inertia of a body, the more difficult it is to stop its rotational motion.

Moment of Inertia of a Single Particle: For a very simple case the moment of inertia of a single particle about an axis is given by, I = mr²Here, m is the mass of the particle and r its distance from the axis under consideration.

Moment of Inertia of a System of Particles

The moment of inertia of a system of particles about an axis is given by :

$I=\sum _i{m}_i{r}_i^2$

Where r_i is the perpendicular distance from the axis to the ith particle, which has a mass m_i.

Moment of Inertia of Rigid Bodies: For a continuous mass distribution such as found in a rigid body, we replace the summation of Eq. (ii) by an integral. If the system is divided into

infinitesimal elements of mass dm and if r is the distance from a mass element to the axis of rotation, the moment of inertia is,

$I=\int r^{2}dm$

Where the integral is taken over the system.

Rigid Body : A rigid body is a body with a definite & ......

Radius of Gyration: Radius of gyration may be defined as the distance from the axis at which, if the whole mass of the body were to be concentrated, the moment of inertia would be the same about the given axis as with its actual distribution of mass.

Suppose a rigid body consists of n particles of each of the mass m. Let r₁, r₂, ...... r_n be the perpendicular distances of these particles from the axis of rotation. Then

$I=m{r}_1^2+m{r}_2^2+.....+m{r}_n^2$

$=m\left[r_1^2+r_2^2+.....+r_n^2\right]$ $=m\times n\left[\frac{r_1^2+r_2^2+.....+r_n^2}{n}\right]$

$=M\left[\frac{r_1^2+r_2^2+.....+r_n^2}{n}\right]$ (where M = m x n) … (i)

If whole mass of the body is regarded to be concentrated at a perpendicular distance K (radius of gyration), then

I = M K² … (ii)

From eqs. (1) and (2), $K={\left[\frac{r_1^2+r_2^2+.....+r_n^2}{n}\right]}^{\frac{1}{2}}$ … (iii)

Therefore, radius of gyration of a body about an axis is equal to the root mean square distance of the constituent particles from the given axis.

Theorem of Perpendicular Axis

Illustration 1: The moment of inertia of a thin square plate ABCD (as shown in figure)

of uniform thickness about an axis passing through the centre O and perpendicular to the Plane of plate is:

(A) I₁ + I₂ (B) I₃ + I₄

(C) I₁ + I₃ (D) I₁ + I₂ + I₃ + I₄

Where I₁, I₂, I₃ and I₄ are moments of inertia about axis 1, 2, 3 and 4 respectively which are in the plane of plate?

Solution: If I₀ is moment of inertia of plane passing through the centre and $\perp$ to plate, then according to the theorem of perpendicular axes.

I₀ = I₁ + I₂ = I₃ + I₄

From symmetry, I₁ = I₂ and I₃ = I₄

$\therefore$ I₀ = 2I₂ = 2I₃ i.e., I₂ = I₃

$\therefore$ I₀ = I₁ + I₂, I₀ = I₃ + I₄

And I₀ = I₁ + I₃ i.e., first three answers are correct. Ans. (A, B, C)

Theorem of Parallel Axes

The moment of inertia of a body about any axis is equal to its moment of inertia about a parallel axis through its centre of gravity plus the product of the mass of the body and the square of the perpendicular distance between the two parallel axes.

Illustration 2: The moment of inertia of a ring about one of its diameters is I. What will be its moment of inertia about a tangent parallel to the diameter?

(A) 4I (B) 2I (C) $\frac{3}{2}I$ (D) 3I

Solution: (D) $I=\frac{1}{2}M{R}^2$. According to theorem of parallel axes.

$I^{\prime }=\frac{1}{2}M{R}^2+M{R}^2=\frac{3}{2}M{R}^2=3I.$