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Chapter 5: Q8P (page 268)

For the solid in Problem 7, Find $I_{Z}$ I Mif $伪 = 蟿蟿 / 3$ and the density is constant.

Short Answer

Expert verified

The required value of $\frac{I_{z}}{M} i s \frac{a^{2}}{4}$ .

Step by step solution

01

Given information

The value of $伪$ is $\frac{蟿蟿}{3}$ and the density is constant.

02

Concept of spherical coordinates

The spherical coordinatesare related to Cartesian coordinates (x,y,z) by:

$r = \sqrt{x^{2} + y^{2} + z^{2} 胃} = \tan^{- 1} (\frac{y}{x}) 蠒 = \cos^{- 1} (\frac{z}{r})$

The cylindrical coordinates are related to Cartesian coordinates (x,y,z)by:

$r = \sqrt{x^{2} + y^{2} 胃} = t a n^{- 1} (\frac{y}{x}) z = z$

03

The moment of inertia

The moment of inertia is calculated as:

$V = \frac{2 蟿蟿}{3} a^{3} (1 - \cos 伪) = \frac{蟿蟿 a^{3}}{3}$

Calculate further as follows:

$I_{z} = 鈭� 蚁 (x^{2} + y^{2}) d V = 蚁 (r^{2} \sin^{2} 胃) d V = 蚁 {鈭�}_{0}^{2 蟿蟿} d 蠒 {鈭�}_{0}^{蟿蟿 / 3} \sin^{3} 胃 d 胃 {鈭�}_{0}^{蟿蟿 / 3} r^{4} d r = - \frac{2 蟿蟿蚁 a^{5}}{5} {鈭�}_{0}^{蟿蟿} \sin^{3} 胃 d 胃$

The above integral can be written as follows:

$I_{z} = - \frac{2 蟿蟿蚁 a^{5}}{5} {鈭�}_{0}^{蟺 / 3} (1 - \cos^{2} 胃) d (\cos 胃) = - \frac{2 蟿蟿蚁 a^{5}}{5} {(\cos 胃 - \frac{\cos^{3} 胃}{3})}_{0}^{蟺 / 3} = \frac{蟺}{12} 蚁 a^{5}$

Substitute localid="1659146809219" $\frac{蟿蟿}{12} 蚁 a^{5}$ for $I_{z}$ and $蚁 \frac{蟿蟿 a^{3}}{3}$ for M in $\frac{I_{z}}{M}$ .

$\frac{I_{z}}{M} = \frac{a^{2}}{4}$

Therefore, the required value of $\frac{I_{z}}{M}$ is $\frac{a^{2}}{4}$ .

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Most popular questions from this chapter

Prove the following two theorems of Pappus: Use Problems 12 and 13 to find the volume and surface area of a torus (doughnut).

In the integral

$I = {鈭�}_{0}^{鈭�} {鈭�}_{0}^{鈭�} \frac{x^{2} + y^{2}}{{(x^{2} + y^{2})}^{2}} e^{- 2 xy} dxdy$ .

Make the change of variables

$u = x^{2} - y^{2} v = 2 xy$

And evaluate I. Hint: Use (4.8) and the accompanying discussion.

Express the integral $I = {鈭�}_{0}^{1} d x {鈭�}_{0}^{\sqrt{1 - x^{2}}} e^{- x^{2} - y^{2}} d y$ as an integral in polar coordinates $(r, 胃)$ and so evaluate it.

$鈭� {(9 + 2 y^{2})}^{- 1}$ over the quadrilateral with vertice $(1, 3), (3, 3), (2, 6), (6, 6)$

In the problems of this section, set up and evaluate the integrals by hand and check your results by computer ${鈭�}_{x = 1}^{2} {鈭�}_{y = 2}^{1} 8 xy dxdy .$

See all solutions

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