Moment of Inertia of Discrete Masses

The moment of inertia, often denoted by $I$, is a quantity expressing an object's tendency to resist angular acceleration. It is the rotational equivalent of mass in linear motion. For discrete masses, the moment of inertia depends on the distribution of the mass in relation to the axis of rotation.

Definition

For a system of discrete masses, the moment of inertia about a given axis is calculated by summing the products of each mass and the square of its perpendicular distance from the axis of rotation. Mathematically, it is expressed as:

$$ I = \sum_{i=1}^{n} m_i r_i^2 $$

where:

• $I$ is the moment of inertia,
• $m_i$ is the mass of the $i^{th}$ particle,
• $r_i$ is the perpendicular distance of the $i^{th}$ particle from the axis of rotation,
• $n$ is the number of discrete masses.

Important Points

• The moment of inertia is a scalar quantity.
• It has units of mass times distance squared (e.g., kg·m²).
• The larger the moment of inertia, the harder it is to change the rotational motion of the object.
• The moment of inertia depends not only on the total mass but also on the distribution of mass relative to the axis of rotation.

Calculation of Moment of Inertia for Common Shapes

For some common shapes and mass distributions, the moment of inertia can be calculated using standard formulas. Here are a few examples:

Shape	Axis of Rotation	Moment of Inertia Formula
Point Mass	Any axis through the center	$I = m r^2$
Rod (length $L$)	Perpendicular to rod through center	$I = \frac{1}{12} m L^2$
Rod (length $L$)	Perpendicular to rod through end	$I = \frac{1}{3} m L^2$
Solid Sphere (radius $R$)	Through center	$I = \frac{2}{5} m R^2$
Hollow Sphere (radius $R$)	Through center	$I = \frac{2}{3} m R^2$
Solid Cylinder (radius $R$, height $h$)	Through center along height	$I = \frac{1}{2} m R^2$
Hollow Cylinder (inner radius $R_1$, outer radius $R_2$, height $h$)	Through center along height	$I = \frac{1}{2} m (R_1^2 + R_2^2)$

Examples

Example 1: Two Point Masses

Consider two point masses, $m_1$ and $m_2$, located at distances $r_1$ and $r_2$ from the axis of rotation, respectively. The moment of inertia of this system is:

$$ I = m_1 r_1^2 + m_2 r_2^2 $$

Example 2: Four Masses in a Square Configuration

Imagine four masses, each of mass $m$, located at the corners of a square with side length $a$. The axis of rotation passes through the center of the square and is perpendicular to the plane of the square. The moment of inertia is:

$$ I = 4m \left(\frac{a}{\sqrt{2}}\right)^2 = 2ma^2 $$

Here, the distance from each mass to the axis of rotation is the diagonal of the square divided by $\sqrt{2}$.

Example 3: Rod with Discrete Masses

Consider a rod of length $L$ with $n$ discrete masses $m_i$ located at distances $r_i$ from one end of the rod. The moment of inertia about an axis perpendicular to the rod and passing through its end is:

$$ I = \sum_{i=1}^{n} m_i r_i^2 $$

Conclusion

The moment of inertia is a fundamental concept in rotational dynamics. It is crucial for understanding how the distribution of mass affects the rotational properties of a system. When dealing with discrete masses, the moment of inertia can be calculated by summing the contributions of each mass, taking into account their distances from the axis of rotation. This concept is not only important in physics but also in engineering, where it is used to design objects that rotate, such as wheels, gears, and turbines.