Center of mass and moments of inertia A solid "trough" of constant density is bounded below by t

Center of mass and moments of inertia A solid "trough" of...

Key Concepts

Triple Integration for Volume Properties

Triple integration is the mathematical tool used to sum physical quantities such as mass, center of mass, and moments of inertia over a three-dimensional region. It involves setting up integrals based on the boundaries of the region and choosing an appropriate order of integration. This technique is essential for calculating properties that depend on the volume distribution of the material.

Center of Mass

The center of mass of a body is the point where the total mass of the body can be considered to be concentrated. For a body with constant density, the calculation simplifies since the density cancels out when computing the average position. The coordinates of the center of mass are determined by integrating the coordinate values over the entire volume and then normalizing by the total volume or mass.

Moments of Inertia

The moment of inertia quantifies the resistance of a body to rotational motion about a specified axis. It is computed by integrating the square of the distance from each infinitesimal mass element to the axis of rotation over the entire volume. This definition extends to any axis in three-dimensional space and is fundamental in analyzing rotational dynamics.

Symmetry in Calculations

Symmetry in the geometric or mass distribution of an object can greatly simplify the computation of physical properties. When a body exhibits symmetry, certain integrals will evaluate to zero or yield equal contributions in different directions, allowing for a reduction in the complexity of the calculations for both the center of mass and the moments of inertia.

Transcript

00:01 Okay, so we are given to find the center of mass and moment of inertia bounded by surface at top by z equals to 4.

00:08 So we have a surface this, z equals to 4, so this plane, right? then we have bottom by z equals to 4 y square.

00:18 So this is this parabolic curve here.

00:23 So this is the surface.

00:25 And then from the front end is x equals to 1.

00:29 So this plane.

00:30 And at bottom, sorry, at rare, it is x minus 1, so this plane.

00:35 Okay, so we need to find the center of mass and moment of inertia of this body.

00:41 Okay, so now if you look into this body, it is symmetric about y -axis, right? that is, if we see here, so every point is symmetric about this.

00:54 So if we take this line, it is symmetric here.

00:56 So the center of mass in y direction that is centered at y of y axis, sorry, will be 0.

01:06 Similarly, if we look into this, it is going from x equals to 1 to minus 1, hence it also symmetric in the x -axis.

01:15 So x -bar is 0 and only we need to find is z -bar.

01:20 So z -bar will be given by or is given by equation 1 over a integration, let's say we are integrating from z 1a to zb and here we'll have z x times f of z d z now since this is stacking of the same planar area this over minus 1 to 1 so we can find first area of this piece which is y and here we have our z axis this is z equals to 4 and this is z equals to 4 y square or we can write y as root z over 2 for this part from here to here and y is minus root z over 2 for this part okay so let's calculate area first area will be region integration d y d z right now y is ranging from minus z over 2 to z over 2 and z ranging from 0 to 4 so this is 0 to 4 the integration of d y is y and this will give us so on solving this we get our area is equal to sorry this is root z right so on calculating we get 24 by okay so now we can calculate this value so here z bar will be 1 by a which is this value now z is starting from again 0 to 4, z is z and f of z that we have already found here.

03:14 It is root z over 2, dz.

03:22 And here, since we are taking this value, so it is only considering this only part, so it will be double to consider this part as well.

03:33 Okay, so now we have two times one over...

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