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Chapter 16: Problem 6

In the integral for the moment \(M_{x z}\) with respect to the \(x z\) -plane of a solid, why does \(y\) appear in the integrand?

Short Answer

Expert verified
Answer: The variable \(y\) appears in the integrand for the moment \(M_{xz}\) because it represents the perpendicular distance from the \(xz\)-plane to the solid's mass at the point \((x, y, z)\). This distance, multiplied by the density of the solid at that point, is integrated over the entire volume of the solid to calculate the moment \(M_{xz}\).

Step by step solution

01

Understanding moments

Moments are a measure of the distribution of mass within a solid. In this case, we are considering the moment \(M_{xz}\), which is the moment of the solid with respect to the \(xz\)-plane.
02

Definition of the moment \(M_{xz}\)

The moment \(M_{xz}\) is defined as the integral of the product of the distance from the \(xz\)-plane and the density of the solid at that point. Mathematically, this is written as: $$ M_{xz} = \iiint_V y \rho(x, y, z) \,dV $$ where \(\rho(x, y, z)\) is the density of the solid at point \((x, y, z)\) and \(dV\) is an infinitesimally small volume element.
03

Explain the appearance of \(y\) in the integrand

In this integral, \(y\) appears because it represents the perpendicular distance from the \(xz\)-plane to the solid's mass at the point \((x, y, z)\). When integrating over the entire volume \(V\) of the solid, we are summing up the product of the density and distance for each infinitesimally small volume element, thereby calculating the moment \(M_{xz}\). That's why \(y\) is in the integrand, as it represents the distance to the \(xz\)-plane.

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