Moment of Inertia of Rectangular Section is explained and Equation is derived for the Centroidal Axis and Its Base in the with following timestamps:
0:00 – Mechanics of Solid Lecture Series
0:11 – Outlines on the Session
0:34 – Moment of Inertia of Rectangular Section about Centroidal Axis
4:32 – Moment of Inertia of Rectangular Section about its base
Following points are covered in this video:
1. Moment of Inertia of Rectangular Section about Centroidal Axis
2. Moment of Inertia of Rectangular Section about its base
Engineering Funda channel is all about Engineering and Technology. Here this video is a part of Mechanics of Solids or Engineering Mechanics.
#MomentofInertia, #RectangularSection, #CentroidalAxis, #EngineeringFunda, #MechanicsOfSolids
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.
Details of Moment of Inertia of Rectangular Section:
The moment of inertia of a rectangular section with respect to an axis passing through its centroid and perpendicular to its width is given by the formula:
I = (1/12) * b * h³
where I is the moment of inertia, b is the width of the rectangular section, and h is the height of the rectangular section.
The centroid of a rectangular section is located at the center of the section and is equidistant from its top and bottom edges and from its left and right edges. The moment of inertia of the rectangular section can be calculated by dividing it into infinitesimal strips of width dx and integrating their contributions to the moment of inertia. For a rectangular section, the moment of inertia is symmetric about both the x-axis and the y-axis, so the contributions from each strip are the same.
Using the parallel axis theorem, the moment of inertia of a rectangular section with respect to an axis passing through its centroid and parallel to its width is given by the formula:
I = (1/12) * b * h³ + (b * h) * (d/2)²
where d is the distance from the centroid of the rectangular section to the axis of rotation. The first term in the equation represents the moment of inertia of the rectangular section about its centroidal axis, while the second term represents the additional moment of inertia due to the offset of the axis of rotation from the centroid.
The moment of inertia of a rectangular section is an important parameter in the design of beams and other structures that are subject to bending loads. By knowing the moment of inertia of a rectangular section, it is possible to calculate the maximum bending stress that the section can withstand and to design structures that will resist bending without failing.