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Magazine Chapter 6: Problem 18
Find the moment of inertia around the \(y\) axis for the ball \(x^{2}+y^{2}+z^{2} \leq R^{2}\) if the mass density is a constant \(\delta\).
Short Answer
Expert verified
The moment of inertia around the y-axis is \(\frac{8}{15} \pi \delta R^5\).
Step by step solution
01
Understand the Geometry and Setup
The problem asks to find the moment of inertia around the y-axis of a solid sphere defined by the equation \(x^2 + y^2 + z^2 \leq R^2\). The moment of inertia \(I_y\) measures how the mass is distributed with respect to the y-axis within the sphere of radius \(R\). The mass density \(\delta\) is constant throughout the volume of the sphere.
02
Define the Volume Element
For a solid sphere, it is effective to use spherical coordinates \((\rho, \theta, \phi)\) where the relationship between Cartesian and spherical coordinates is given by: - \( x = \rho \sin\phi \cos\theta \)- \( y = \rho \sin\phi \sin\theta \)- \( z = \rho \cos\phi \)The differential volume element in spherical coordinates is \(dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta\).
03
Moment of Inertia Formula
The moment of inertia \( I_y \) about the y-axis is given by the integral\[ I_y = \int_V (x^2 + z^2) \, dm \]where \( dm = \delta \, dV \). Here, \( x^2 + z^2 = \rho^2 \sin^2\phi \cos^2\theta + \rho^2 \cos^2\phi \) in spherical coordinates.
04
Set Up the Integral
Substitute \(dm\) and the expression for \(x^2 + z^2\) into the moment of inertia formula:\[ I_y = \int_0^{2\pi} \int_0^{\pi} \int_0^R \delta (\rho^2 \sin^2\phi \cos^2\theta + \rho^2 \cos^2\phi) \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \]
05
Evaluate the Integral with Respect to \(\theta\)
Perform the integration over \( \theta \):\[ \int_0^{2\pi} \cos^2\theta \, d\theta = \pi \]Since the integral of the second part is 0 over one full period \(2\pi\), only the \(\cos^2\theta\) part contributes to the \(\theta\) integral.
06
Simplify and Calculate Remaining Integrals
Simplify and complete the integrals over \( \phi \) and \( \rho \):- Integral over \( \phi \): \[ \int_0^{\pi} \sin^3\phi \, d\phi = \frac{4}{3} \]- Integral over \( \rho \): \[ \int_0^{R} \rho^4 \, d\rho = \frac{R^5}{5} \]Substitute these results back into the integral equations.
07
Calculate the Moment of Inertia
Combining all integrated results yields:\[ I_y = \delta \cdot \pi \cdot \frac{4}{3} \cdot \frac{R^5}{5}\]Simplify to get the moment of inertia:\[ I_y = \frac{8}{15} \pi \delta R^5 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Spherical Coordinates
Spherical coordinates are a system of three numbers \( \rho, \theta, \phi \) used to define a point in three-dimensional space. They are particularly useful for problems with symmetry around a point, like spheres.
- \( \rho \) is the distance from the origin to the point.
- \( \phi \) is the angle between the positive z-axis and the line formed by the origin and the point. It ranges from 0 to \pi.
- \( \theta \) is the angle between the positive x-axis and the projection of the line on the xy-plane, ranging from 0 to 2\pi.
- \( x = \rho \, \sin\phi \, \cos\theta \)
- \( y = \rho \, \sin\phi \, \sin\theta \)
- \( z = \rho \, \cos\phi \)
Mass Density
Mass density, often denoted as \( \delta \), is a measure of mass per unit volume. For the solid sphere problem, we assume a constant mass density, meaning that the sphere's mass is evenly distributed throughout its volume.
Understanding mass density provides insight into determining how a shape's composition affects its inertia and other physical properties.
- In simpler terms, if you look at any small part of the sphere, it has the same amount of mass as any other similarly-sized part.
Understanding mass density provides insight into determining how a shape's composition affects its inertia and other physical properties.
Double Integration
Double integration is a technique used to compute areas or volumes of regions, particularly those that can be described in two-dimensional integrals. When extended to triple integrals, like in this sphere problem, it can help calculate volume, mass, and moment of inertia in three-dimensional space.
To set up a double or triple integral:
To set up a double or triple integral:
- Identify the limits of integration. For the spherical coordinates, \( \theta \) spans from 0 to 2\pi, and \( \phi \) from 0 to \pi.
- Define the integrand, the function to be integrated. In this case, it relates to the moment of inertia \( I_y \).
- Evaluate the integrals in sequence, typically starting with the innermost integral and moving outward.
Solid Sphere
A solid sphere is a perfectly symmetrical three-dimensional object where every point on its surface is equidistant from its center. The symmetry of a solid sphere simplifies many calculations in physics and engineering because its properties are uniform in all directions.
When dealing with the moment of inertia for solid spheres:
When dealing with the moment of inertia for solid spheres:
- We utilize spherical symmetry, making it easier to resolve integrations in spherical coordinates.
- The sphere's uniform density means all parts contribute equally to its mass-based properties.
- Calculations like those for the moment of inertia benefit from this symmetry, leading to elegant solutions.
