In physics, a moment of inertia is a quantitative measure of a body’s rotational inertia—that is, the resistance that the body exhibits to having its speed of rotation about an axis shifted as a result of torque application (turning force) . The axis can be internal or external, as well as fixed or not. However, the moment of inertia (I) is always specified in relation to that axis and is defined as the sum of the products obtained by multiplying the mass of each particle of matter in a given body by the square of the distance from the axis. When calculating angular momentum for a rigid body, the moment of inertia is analogous to mass in linear momentum.
The linear momentum p is equal to the mass m multiplied by the velocity v, whereas the angular momentum L is equal to the moment of inertia I multiplied by the angular velocity.
The moment of inertia unit is a composite unit of measurement. In the International System (SI), m is measured in kilogrammes and r is measured in metres, with I (moment of inertia) having the dimension kilogram-metre square. In the United States, m is measured in slugs (1 slug = 32.2 pounds) and r is measured in feet, with I expressed in terms of the slug-foot square.
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The moment of inertia is defined as the quantity expressed by the body resisting angular acceleration, which is the sum of the product of each particle’s mass and its square of the distance from the axis of rotation. In simpler terms, it is a quantity that determines the amount of torque required for a given angular acceleration in a rotational axis. The moment of inertia is also known as the angular mass or rotational inertia. The SI unit for the moment of inertia is kg m2.
In general, Moment of Inertia is written as I = m × r2
where,
m = The sum of the mass products.
r = Distance from the rotation’s axis.
as well as the integral form:I = ∫dI = ∫0M r2 dm
M1 L2 T0 is the dimensional formula for the moment of inertia.
The moment of inertia, like mass, plays the same role in linear motion. It is a measurement of the resistance of a body to a change in rotational motion. It is constant for a given rigid frame and rotation axis.
Moment of inertia, I = ∑mi ri2. . . . . . . (1)
Kinetic Energy, K = ½ I ω2 . . . . . . . . . (2)
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Parallel Axis Theorem
If the moment of inertia about the center of mass is , the MOI about an axis parallel to it and at a distance is:
Where is the total mass of the object.
Perpendicular Axis Theorem
For a planar object, the moment of inertia about an axis perpendicular to the plane (-axis) is the sum of the MOI about two perpendicular axes in the plane (-axis and -axis):
The moment of inertia is affected by the following factors:
The moment of inertia quantifies how the mass of an object is distributed relative to the axis of rotation. Mathematically, for a system of particles:
For a continuous distribution of mass:
Where:
The resistance to angular acceleration is defined as the moment of inertia (i.e. distribution of mass). Inertia is a linear property of mass that describes how much an object resists a change in motion caused by a force, or how massive it is.
The moment of inertia is a measure of an object's resistance to changes in its rotational motion about a specific axis. It depends on the mass of the object and how that mass is distributed relative to the axis of rotation.
The MOI is specific to the axis of rotation. Shifting the axis changes the values (distance from the axis), which directly impacts the M
