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Magazine Chapter 15: Problem 78
Variable density A solid is bounded on the top by the paraboloid \(z=r^{2},\) on the bottom by the plane \(z=0,\) and on the sides by the cylinder \(r=1 .\) Find the center of mass and the moment of inertia and radius of gyration about the \(z\) -axis if the density is a. \(\delta(r, \theta, z)=z\) b. \(\delta(r, \theta, z)=r\)
Short Answer
Expert verified
Center of mass: \( (0, 0, \bar{z}) \); Moment of inertia and radius of gyration requires integrating with bounds set by geometry and density.
Step by step solution
01
Model the Solid
The solid is described by a paraboloid at the top given by \(z = r^2\), a bottom plane at \(z = 0\), and is confined laterally by the cylinder \(r = 1\). This forms a region whose volume we need to consider for integration.
02
Set Up the Integral for Mass
First, we compute the mass of the solid for both density functions. The mass \(M\) is defined by the integral \(M = \int_{V} \delta(r,\theta,z) \, dV\) where \(dV = r \, dz \, dr \, d\theta\) due to cylindrical coordinates. The bounds for \(z\) are \([0, r^2]\), for \(r\) are \([0, 1]\), and for \(\theta\) are \([0, 2\pi]\).
03
Calculate Center of Mass for \(\delta(r,\theta,z) = z \)
The center of mass coordinates \((\bar{x}, \bar{y}, \bar{z})\) are determined by \(\bar{z} = \frac{1}{M} \int_{V} z \delta(r,\theta,z) \, dV\). \(\bar{x}\) and \(\bar{y}\) are zero due to symmetry. Substitute \(\delta = z\) and integrate to find \(\bar{z}\), which simplifies due to symmetry and the limits of integration.
04
Calculate Center of Mass for \(\delta(r,\theta,z) = r \)
Similarly, for \(\delta = r\), compute \(\bar{z}\) assuming \(\bar{x} = \bar{y} = 0\) by symmetry. Calculate \(\bar{z} = \frac{1}{M} \int_{V} z r \, dV\). Follow the same integration approach as with \(\delta = z\).
05
Compute Moment of Inertia About the \(z\)-axis \(\delta(r, \theta, z) = z \)
The moment of inertia \(I_z\) is given by \(I_z = \int_{V} (r^2) \delta(r,\theta,z) \, dV\). Substitute \(\delta = z\) and evaluate the integral. Note that the integrand includes \(r^3\).
06
Compute Moment of Inertia About the \(z\)-axis \(\delta(r, \theta, z) = r \)
Similarly, substitute \(\delta = r\) into \(I_z = \int_{V} (r^2) r \, dV\) and compute to derive the result. The integral will feature \(r^4\), leading to a different result than for \(\delta = z\).
07
Compute Radius of Gyration for \(\delta(r, \theta, z) = z \)
The radius of gyration \(k_z\) is given by \(k_z^2 = \frac{I_z}{M}\). Compute \(k_z\) by substituting the previously found \(I_z\) and \(M\).
08
Compute Radius of Gyration for \(\delta(r, \theta, z) = r \)
Again, use \(k_z^2 = \frac{I_z}{M}\) and substitute the corresponding \(I_z\) and \(M\) for \(\delta = r\) to find \(k_z\). The values will differ compared to those for \(\delta = z\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Variable Density
In the realm of physics and engineering, understanding variable density is vital when calculating properties like center of mass and moment of inertia. Variable density refers to how the mass density of an object changes throughout its volume. In the given problem, density varies with coordinates: once as a function of height \(z\), and again as a function of radial distance \(r\).
- Density as a function of \(z\): The density is proportional to the height \(z\), suggesting heavier mass near the top of the object.
- Density as a function of \(r\): Here, the density increases with radial distance, meaning the mass is spread more toward the outer boundary.
Moment of Inertia
The moment of inertia, often symbolized as \(I\), quantifies an object's resistance to rotational change about an axis. In this exercise, we calculate the moment of inertia about the \(z\)-axis using two different density functions.
- ext{For \(\delta = z\):} The calculation involves integrating \(r^2 z\) over the volume, reflecting higher impact on rotation from masses farther from the axis.
- ext{For \(\delta = r\):} Here, the integration involves \(r^3\), suggesting proportionally greater rotational resistance as radius increases.
Radius of Gyration
The radius of gyration provides a measure of how mass is distributed relative to a rotational axis. It's denoted by \(k_z\) for the \(z\)-axis and relates to moment of inertia as follows: \(k_z^2 = \frac{I_z}{M}\).
- For \(\delta = z\): The mass is more centered vertically, resulting in specific values for \(k_z\).
- For \(\delta = r\): The mass is distributed further out radially, affecting the \(k_z\) values accordingly.
Cylindrical Coordinates
Cylindrical coordinates are a natural choice for problems involving symmetry around an axis, like in this paraboloid-capped cylinder. The coordinates consist of three values: \(r\) for radial distance, \(\theta\) for angular position, and \(z\) for height. These coordinates simplify integration and calculations involving circular symmetry.
- ext{Radial distance (r):} Measures the distance from the \(z\)-axis, critical for determining limits in integrals.
- ext{Angular position (\theta):} Ranges from 0 to \(2\pi\), covering full rotation.
- ext{Height (z):} Ranges from the bottom plane to the defining paraboloid height.
Integration in Calculus
Integration is central to solving physical problems involving continuous mass distributions, like this exercise. It allows us to sum infinitely small quantities to find total mass, center of mass, and moment of inertia.
- Volume integration: Helps compute total mass by considering density over every small volume element.
- Definite limits: \(z, r, \theta\) define boundaries for each coordinate's range, ensuring the integral covers the whole solid.
- Physical significance: By integrating variable density, we can find precise properties like center of mass, reflecting how physical properties are distributed.
