Examples based on Moment of Inertia is solved with following timestamps:

0:00 – Mechanics of Solid Lecture series
0:09 – Outlines on the session
0:16 – Examples based on Centroid of Composite Linear Elements

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Examples based on Moment of Inertia

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Details of Moment of Inertia:

The moment of inertia, also known as rotational inertia, is a measure of an object's resistance to rotational motion. It is defined as the sum of the products of the mass of each particle in an object and the square of its distance from a given axis of rotation. The moment of inertia depends not only on the mass of the object but also on the distribution of the mass around the axis of rotation.

The moment of inertia is denoted by the symbol I and has units of kg m² or m⁴. The moment of inertia is often used in the analysis of rotating objects, such as wheels, gears, and flywheels.

The moment of inertia can be calculated using the following integral:

I = ∫ r² dm

where I is the moment of inertia, r is the perpendicular distance from the particle to the axis of rotation, and dm is an infinitesimal mass element.

For simple shapes, there are known formulas for the moment of inertia. For example, the moment of inertia of a rectangular plate of mass M, length L, and width W, rotating about an axis perpendicular to its length and passing through its center, is:

I = (1/12) * M * (L² + W²)

The moment of inertia of a solid cylinder of mass M, radius R, and length L, rotating about its central axis, is:

I = (1/2) * M * R²

The moment of inertia of a hollow cylinder of mass M, inner radius R1, outer radius R2, and length L, rotating about its central axis, is:

I = (1/2) * M * (R2² + R1²)

The moment of inertia of a solid sphere of mass M and radius R, rotating about an axis passing through its center, is:

I = (2/5) * M * R²

By calculating the moment of inertia of an object, it is possible to predict its response to various types of rotational motion, such as spinning, rolling, and tipping over.