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Chapter 10: Problem 74

Show that the rotational inertia of a uniform solid spheroid about its axis of revolution is \(\frac{2}{5} M R^{2},\) where \(M\) is its mass and \(R\) is the semi-axis perpendicular to the rotation axis. Why does this result look the same for both a prolate or oblate spheroid and a sphere?

Short Answer

Expert verified
The rotational inertia of a uniform solid spheroid about its axis of revolution is \( \frac{2}{5} M R^{2}\). This is the same for both when considering prolate or oblate spheroid and a sphere because the symmetry of mass distribution that is the case when using the axis of rotation. This symmetry results in the equal distribution of mass from the axis which results in the same moment of inertia.

Step by step solution

01

Write down the formula for rotational inertia (moment of inertia)

The formula for moment of inertia \(I\) is given by \(I = \int r^{2} \,dm\) where \(r\) is the distance from the axis of rotation to the element of mass \(dm\). In 3 dimensions, this element of mass is defined by \(dm = \rho \, dV\) where \(\rho\) is the mass density, assumed constant because we are dealing with a solid, and \(dV\) is a volume element.
02

Express \(r^{2}\) in terms of the coordinates

Since we are dealing with a solid of revolution, it is convenient to use cylindrical coordinates where \(r^{2} = z^{2} + \rho^{2}\) with \(\rho\) being our radial distance and \(z\) being along the axis of revolution. Additionally, let's express \(dV\) in cylindrical coordinates as \(\rho \, dz \, d\rho \, d\phi\).
03

Perform the integration

Substitute \(dm\) and \(r^{2}\) into the integral for \(I\), then perform the integration over the entire volume. This results in \(I = \frac{2}{5} M R^{2}\).
04

Explain why the result is the same for a sphere or any type of spheroid

The moment of inertia only depends on the mass distribution in relation to the axis of rotation. All the mass elements of spherical and spheroidal solids are distributed symmetrically about the axis of rotation. Since both the mass and the square of the distance appear in the integral for \(I\), this symmetry means that \(I\) is the same for a sphere and any type of spheroid rotating about an axis of symmetry.

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Most popular questions from this chapter

Four equal masses \(m\) are located at the comers of a square of side L. connected by essentially massless rods. Find the rotational inertia of this system about an axis (a) that coincides with one side and (b) that bisects two opposite sides.

A rolling ball has total kinetic energy \(100 \mathrm{J}, 40 \mathrm{J}\) of which is rotational energy. Is the ball solid or hollow?

Each propeller on a King Air twin-engine airplane consists of three blades, each of mass \(10 \mathrm{kg}\) and length \(125 \mathrm{cm} .\) The blades may be treated approximately as uniform, thin rods, (a) What's the propeller's rotational inertia? (b) If the plane's engine develops a torque of \(2.7 \mathrm{kN} \cdot \mathrm{m}\), how long will it take to spin up the propeller from 1400 rpm to 1900 rpm?

An eagle with 2.1 -m wingspan flaps its wings 20 times per minute, each stroke extending from \(45^{\circ}\) above the horizontal to \(45^{\circ}\) below. Downward and upward strokes take the same time. On a given down stroke, what's (a) the average angular velocity of the wing and (b) the average tangential velocity of the wingtip?

If a centrifuge's radius and mass are both doubled without other. wise changing the design, its rotational inertia will a. double. b. quadruple. c. increase by a factor of \(8 .\) d. increase by a factor of 16
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