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Magazine Chapter 14: Problem 20
Find the center of gravity of an object that occupies the given region and has the given mass density. The solid region bounded above by the sphere \(x^{2}+y^{2}+\) \(z^{2}=4\) and below by the upper nappe of the cone \(z^{2}=\) \(x^{2}+y^{2} ; \delta(x, y, z)=z^{2}\left(x^{2}+y^{2}+z^{2}\right)\)
Short Answer
Expert verified
The center of gravity is at (0, 0, \(\bar{z}\)).
Step by step solution
01
Cylindrical Coordinates Setup
Convert the spherical and conical equations into cylindrical coordinates. Recall that in cylindrical coordinates, \(x = r \cos \theta\), \(y = r \sin \theta\), and \(z = z\). - The sphere equation \(x^2 + y^2 + z^2 = 4\) becomes \(r^2 + z^2 = 4\).- The cone equation \(z^2 = x^2 + y^2\) becomes \(z^2 = r^2\), or \(z = \pm r\).The object is the region where \(z\) is between \(r\) (the cone) and \(\sqrt{4-r^2}\) (the sphere).
02
Symmetry Consideration
Consider the symmetry of the solid region. It is symmetric about the z-axis due to both the sphere and cone equations being functions of \(x^2 + y^2\).Because of this symmetry around the z-axis, the center of gravity will have \(x = 0\) and \(y = 0\). Thus, we only need to find the z-coordinate of the center of gravity (\(\bar{z}\)).
03
Mass Element in Terms of Cylindrical Coordinates
The density function in cylindrical coordinates is given by \(\delta(r, \theta, z) = z^2(r^2 + z^2)\). The volume element in cylindrical coordinates is \(dV = r \, dr \, d\theta \, dz\).So, the mass element is \(dm = \delta \, dV = z^2 (r^2 + z^2) \, r \, dr \, d\theta \, dz\).
04
Evaluate the Total Mass
Integrate the mass element over the entire volume to find the total mass.\[ M = \int_0^{2\pi} \int_0^{\sqrt{2}} \int_r^{\sqrt{4-r^2}} z^2 (r^2 + z^2) r \, dz \, dr \, d\theta \]Perform the integration with respect to \(z\) first, \(r\) next, and finally \(\theta\).
05
Evaluate the First Moment About Z-Axis
Compute the first moment about the z-axis, which involves integrating \(z\cdot dm\):\[ M_z = \int_0^{2\pi} \int_0^{\sqrt{2}} \int_r^{\sqrt{4-r^2}} z^3 (r^2 + z^2) r \, dz \, dr \, d\theta \]Perform the integration following the same order as for the total mass.
06
Calculate Center of Gravity (Z-Coordinate)
The z-coordinate of the center of gravity is given by \(\bar{z} = \frac{M_z}{M}\). Divide the moment computed in Step 5 by the total mass from Step 4. This results in the z-coordinate of the center of gravity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cylindrical Coordinates
Cylindrical coordinates are a three-dimensional coordinate system that is particularly useful for solving problems involving symmetry around a central axis. They combine elements of both polar coordinates and Cartesian coordinates. In cylindrical coordinates:
- The variable \(r\) represents the distance from the z-axis in the xy-plane.
- The angle \(\theta\) is measured from the positive x-axis in the xy-plane.
- The variable \(z\) keeps its usual Cartesian meaning, representing height.
- \(x = r \cos \theta\)
- \(y = r \sin \theta\)
- \(z = z\)
Mass Density
Mass density in mathematical problems is a function that tells us how much mass is present per unit volume of a material body. In this exercise, mass density \(\delta(x, y, z)\) is given as \(z^2(x^2 + y^2 + z^2)\). When working in cylindrical coordinates, it transforms to \(\delta(r, \theta, z) = z^2(r^2 + z^2)\).
By integrating this density over a region, we can determine various properties of the object, such as total mass.
In physical applications:
By integrating this density over a region, we can determine various properties of the object, such as total mass.
In physical applications:
- Higher density in a region signifies more mass concentration there.
- The density function might be dependent on positional variables, typically reflecting spatial variations.
Volume Integration
Volume integration in cylindrical coordinates involves evaluating a triple integral over a specified region. It is essential when we need to calculate properties like mass, center of mass, and moments of inertia. The volume element in cylindrical coordinates is given by:
- \(dV = r \ dr \ d\theta \ dz\)
- To calculate total mass, integrate the mass element \(dm = z^2 (r^2 + z^2) r \ dr \ d\theta \ dz\) over the symmetric region defined by the sphere and cone.
- To find the first moment about the z-axis, integrate \(z \cdot dm\).
Symmetry Consideration
Symmetry consideration simplifies complex integrations by reducing the dimensionality or number of variables. In this problem, the object is symmetric about the z-axis. This symmetry arises because both the sphere \(x^2 + y^2 + z^2 = 4\) and the cone \(z^2 = x^2 + y^2\) depend on \(x^2 + y^2\). So if we were to rotate the object around the z-axis, its appearance would remain unchanged.
- This means the center of gravity in the xy-plane is at the origin, making \(x = 0\) and \(y = 0\).
- Only the calculation for the z-coordinate of the center of gravity is necessary.
