Find study content
Learning Materials

Discover learning materials by subject, university or textbook.

Explanations

Textbooks

Exams
All Subjects

Anthropology

Archaeology

Architecture

Art and Design

Bengali

Biology

Business Studies

Greek

Chemistry

Chinese

Combined Science

Computer Science

Economics

Engineering

English

English Literature

Environmental Science

French

Geography

German

History

Hospitality and Tourism

Human Geography

Italian

Japanese

Law

Macroeconomics

Marketing

Math

Media Studies

Medicine

Microeconomics

Music

Nursing

Nutrition and Food Science

Physics

Polish

Politics

Psychology

Religious Studies

Sociology

Spanish

Sports Sciences

Translation

All Subjects

Biology

Business Studies

Chemistry

Combined Science

Computer Science

Economics

English

Environmental Science

Geography

History

Math

Physics

Psychology

Sociology

EXAM TYPES

GCSE

IGCSE

AS

A Level

International A Level

University Admissions Tests

GCSE SUBJECTS

GCSE 1

GCSE 2

GCSE 3

GCSE 4

GCSE 5

SOME-TEXT

IGCSE SUBJECTS

IGCSE 1

AS SUBJECTS

AS 1

A Level SUBJECTS

A Level 1

International A Level SUBJECTS

International A Level 1

University Admissions Tests SUBJECTS

University Admissions Tests 1
Features
Features

Discover all of these amazing features with a free account.

Flashcards

Vaia AI

Notes

Study Plans

Study Sets
What’s new?

Flashcards
Study your flashcards with three learning modes.

Study Sets
All of your learning materials stored in one place.

Notes
Create and edit notes or documents.

Study Plans
Organise your studies and prepare for exams.
Resources
Discover

All the hacks around your studies and career - in one place.

Find a job

Find a degree

Student Deals

Magazine

Mobile App
Featured

Magazine
Trusted advice for anyone who wants to ace their studies & career.

Job Board
The largest student job board with the most exciting opportunities.

Vaia Deals
Verified student deals from top brands.

Our App
Discover our mobile app to take your studies anywhere.

Go to App

Learning Materials

Features

Discover

Chapter 5: Q46P (page 248)

Find the mass of a cube of side 2 if the density is proportional to the square of the distance from the center of the cube.

Short Answer

Expert verified

The required Mass of a cube is $8K$ .

Step by step solution

01

Definition of Mass

A physical body's mass is the amount of matter it contains. It's also a measure of inertia, or the resistance to acceleration when a net force is applied.

02

Finding the Mass of a cube

The origin should be in the center of the cube. We may compute the mass in one octant and then double the result by 8 because the cube's density is symmetrical in x, y and z .

Now, Find the Mass,

$M = 8 K \int_{0}^{1} (x^{2} + y^{2} + z^{2}) d x \int_{0}^{1} d y \int_{0}^{1} d z = {8 K \int_{0}^{1} d x \int_{0}^{1} d y (x^{2} z + y^{2} z + \frac{1}{3} z^{3})|}_{0}^{1} = 8 K \int_{0}^{1} d x \int_{0}^{1} d y (x^{2} + y^{2} + \frac{1}{3}) = {8 K \int_{0}^{1} d x (x^{2} y + \frac{1}{3} y^{3} + \frac{1}{3} y)|}_{0}^{1} = 8 K \int_{0}^{1} d x (x^{2} + \frac{1}{3} + \frac{1}{3})$

The final value is obtained using further calculation.

$M = 8 K \int_{0}^{1} (x^{2} + \frac{2}{3}) d x = {8 K [\frac{x^{3}}{3} + \frac{2}{3} x]|}_{0}^{1} = 8 K (\frac{1}{3} + \frac{2}{3}) = 8 K$

M = 8 K \int_{0}^{1} (x^{2} + \frac{2}{3}) d x = {8 K [\frac{x^{3}}{3} + \frac{2}{3} x]|}_{0}^{1} = 8 K (\frac{1}{3} + \frac{2}{3}) = 8 K

Therefore, the Mass of a cube is $8 K$ .

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!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Above the triangle with vertices (0,2),(1,1) and (2,2) , and under the surface z = xy.

The volume inside a sphere of radius ris $V = \frac{4}{3} {ττr}^{3}$ . Then $dV = 4 {ττr}^{2} dr = Adr$ whereis the area of the sphere. What is the geometrical meaning of the fact that the derivative of the volume is the area? Could you use this fact to find the volume formula given the area formula?

$\int_{y = 0}^{1} \int_{x = y^{2}}^{1} \frac{e^{x}}{\sqrt{x}} dxdy$

A lamina covering the quarter disk $x^{2} + y^{2} \leq 4, x > 0, y > 0$ has (area) density . Find the mass of the lamina.

Use Problems 12 and 13 to find the centroids of a semi-circular area and of a semi-circular arc. Hint: Assume the formulas $A = 4 π r^{2}$ , $V = \frac{4}{3} π r^{3}$ for a sphere.

See all solutions

Recommended explanations on Physics Textbooks

Fluids

Read Explanation

Turning Points in Physics

Read Explanation

Mechanics and Materials

Read Explanation

Dynamics

Read Explanation

Radiation

Read Explanation

Medical Physics

Read Explanation

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free