Chapter 5: Q6P (page 268)

Find the mass of the solid in Problem 5 if the density is ${(x^{2} + y^{2} + z^{2})}^{- 1}$ . Check your work by doing the problem in both spherical and cylindrical coordinates.

Short Answer

Expert verified

The required mass is $π \ln 2$ .

Step by step solution

Given information

The density of the solid is $(x^{2} + y^{2} + z^{2})^{- 1}$ .

Concept of the spherical coordinates and cylindrical coordinates

The spherical coordinates $(r, θ, ϕ)$ is related to Cartesian coordinates $(x, y, z)$ by:

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

The cylindrical coordinates $(r, θ, z)$ is related to Cartesian coordinates $(x, y, z)$ by:

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

Calculation to find the required mass by the use of density in spherical coordinates

In spherical coordinates the density is $ρ = \frac{1}{r^{2}}$ .

Calculate as follows:

$M = \int_{0}^{2 τ τ} d ϕ \int_{0}^{τ τ / 4} \sin θ d θ \int_{1}^{\frac{2}{\cos θ}} \frac{1}{r^{2}} r^{2} d r a = 2 τ τ \int_{0}^{τ τ / 4} \tan θ d θ = 2 τ τ \int_{0}^{τ τ / 4} \tan θ d θ = τ τ \ln 2$

Calculation to find the required mass by the use of density in cylindrical coordinates

In the cylindrical coordinates the density is $ρ = \frac{1}{z^{2} + r^{2}}$ .

Calculate as follows:

$M = \int_{0}^{2 τ τ} d θ \int_{1}^{2} d z \int_{0}^{z} \frac{r}{z^{2} + r^{2}} r^{2} d r = 2 τ τ \int_{1}^{2} d z \int_{0}^{z} \frac{d (z^{2} + r^{2})}{z^{2} + r^{2}} (\frac{1}{2}) = 2 τ τ \int_{1}^{2} (\ln 2 z^{2} - \ln z^{2}) d z = τ τ \ln 2$

Therefore, the required mass is $τ τ$ in 2.

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Find the mass of the solid in Problem 5 if the density is ${(x^{2} + y^{2} + z^{2})}^{- 1}$ . Check your work by doing the problem in both spherical and cylindrical coordinates.

Short Answer

Step by step solution

Given information

Concept of the spherical coordinates and cylindrical coordinates

Calculation to find the required mass by the use of density in spherical coordinates

Calculation to find the required mass by the use of density in cylindrical coordinates

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Find the mass of the solid in Problem 5 if the density is (x2+y2+z2)-1. Check your work by doing the problem in both spherical and cylindrical coordinates.

Short Answer

Step by step solution

Given information

Concept of the spherical coordinates and cylindrical coordinates

Calculation to find the required mass by the use of density in spherical coordinates

Calculation to find the required mass by the use of density in cylindrical coordinates

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electromagnetism

Mechanics and Materials

Geometrical and Physical Optics

Atoms and Radioactivity

Thermodynamics

Physical Quantities And Units

Study anywhere. Anytime. Across all devices.

Find the mass of the solid in Problem 5 if the density is ${(x^{2} + y^{2} + z^{2})}^{- 1}$ . Check your work by doing the problem in both spherical and cylindrical coordinates.