Chapter 13: Q 19. (page 1066)

Show that the mass of $\in$ is $\frac{1}{6} πk$ by evaluating the integral:
$\int \int \int_{\in} k d V = \int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} \int_{0}^{1} k ρ^{2} \sin ϕ d ρ d θ d ϕ$

Short Answer

Expert verified

Use spherical coordinates $(ρ, θ, ϕ)$ while evaluating $d V$ using triple integral.

Step by step solution

Given Information

Spherical coordinates are $(ρ, θ, ϕ)$ . We need to prove this by solving given integral.

Simplification

Taking LHS.

$m = \int \int \int_{\in} k d V$

$= \int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} \int_{0}^{1} k ρ^{2} \sin ϕ d ρ d θ d ϕ$

$= k \int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} (\int_{0}^{1} ρ^{2} \sin ϕ d ρ) d θ d ϕ$

$= k \int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} {(\frac{ρ^{3}}{3})}^{1}_{0} \sin ϕ d θ d ϕ$

$= k \int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} \frac{1}{3} \sin ϕ d θ d ϕ$

role="math" localid="1652246785701" $= \frac{k}{3} \int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} \sin ϕ d θ d ϕ$

role="math" localid="1652246883750" $= \frac{k}{3} \int_{0}^{\frac{π}{2}} {[θ]}^{\frac{π}{2}}_{0} \sin ϕ d ϕ$

$= \frac{k}{3} \int_{0}^{\frac{π}{2}} [\frac{π}{2} - 0] \sin ϕ d ϕ$

$= \frac{k π}{6} {[- \cos ϕ]}^{\frac{π}{2}}_{0}$

$= - \frac{k π}{6} [\cos \frac{π}{2} - \cos 0]$

$= - \frac{k π}{6} [0 - 1]$

$= \frac{πk}{6}$

Hence, the mass is $\frac{1}{6} πk$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Show that the mass of $\in$ is $\frac{1}{6} πk$ by evaluating the integral:
$\int \int \int_{\in} k d V = \int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} \int_{0}^{1} k ρ^{2} \sin ϕ d ρ d θ d ϕ$

Short Answer

Step by step solution

Given Information

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Pure Maths

Probability and Statistics

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Show that the mass of ∈is 16πkby evaluating the integral:∫∫∫∈kdV=∫0π2∫0π2∫01kρ2sinϕdρdθdϕ

Short Answer

Step by step solution

Given Information

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Pure Maths

Probability and Statistics

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Show that the mass of $\in$ is $\frac{1}{6} πk$ by evaluating the integral:
$\int \int \int_{\in} k d V = \int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} \int_{0}^{1} k ρ^{2} \sin ϕ d ρ d θ d ϕ$