Chapter 5: Q8P (page 268)

For the solid in Problem 7, Find $I_{Z}$ I Mif $α = τ τ / 3$ 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

Given information

The value of $α$ is $\frac{τ τ}{3}$ and the density is constant.

Concept of spherical coordinates

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

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

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

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

The moment of inertia

The moment of inertia is calculated as:

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

Calculate further as follows:

$I_{z} = \int ρ (x^{2} + y^{2}) d V = ρ (r^{2} \sin^{2} θ) d V = ρ \int_{0}^{2 τ τ} d ϕ \int_{0}^{τ τ / 3} \sin^{3} θ d θ \int_{0}^{τ τ / 3} r^{4} d r = - \frac{2 τ τ ρ a^{5}}{5} \int_{0}^{τ τ} \sin^{3} θ d θ$

The above integral can be written as follows:

$I_{z} = - \frac{2 τ τ ρ a^{5}}{5} \int_{0}^{π / 3} (1 - \cos^{2} θ) d (\cos θ) = - \frac{2 τ τ ρ a^{5}}{5} {(\cos θ - \frac{\cos^{3} θ}{3})}_{0}^{π / 3} = \frac{π}{12} ρ a^{5}$

Substitute localid="1659146809219" $\frac{τ τ}{12} ρ a^{5}$ for $I_{z}$ and $ρ \frac{τ τ a^{3}}{3}$ 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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For the solid in Problem 7, Find $I_{Z}$ I Mif $α = τ τ / 3$ and the density is constant.

Short Answer

Step by step solution

Given information

Concept of spherical coordinates

The moment of inertia

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Most popular questions from this chapter

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For the solid in Problem 7, FindIZ I Mifα=ττ/3and the density is constant.

Short Answer

Step by step solution

Given information

Concept of spherical coordinates

The moment of inertia

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Thermodynamics

Mechanics and Materials

Radiation

Fluids

Electromagnetism

Modern Physics

Study anywhere. Anytime. Across all devices.

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