The moment of inertia (I) is a property of an object that describes its resistance to rotational motion around a particular axis. It depends on the mass distribution of the object and how the mass is distributed relative to the axis of rotation.
The moment of inertia is mathematically defined as the sum of the products of mass elements (dm) and their distances squared (r^2) from the axis of rotation:
I = ∫ r^2 dm
The integral represents the summing of infinitesimally small mass elements throughout the object. The moment of inertia is essentially a measure of how the mass is distributed around the axis of rotation.
The moment of inertia can vary depending on the shape and mass distribution of the object. For example, a point mass rotating around an axis at a distance 'r' has a moment of inertia of I = m * r^2, where 'm' is the mass of the point object.
The moment of inertia plays a crucial role in rotational dynamics, as it appears in Newton's second law for rotational motion, which states that the torque (τ) applied to an object is equal to the rate of change of its angular momentum (L):
τ = dL/dt = I * α
Here, α represents the angular acceleration of the object.
In summary, the moment of inertia is a property that characterizes an object's resistance to rotational motion. It depends on the distribution of mass around the axis of rotation and is defined as the sum of the products of mass elements and their distances squared from the axis.