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Math
CalculusMultivariable Calculus8th Edition

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Chapter 13.4, Problem 44E
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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

Solutions

Chapter
Section
Problem 1E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
Problem 9E
Problem 10E
Problem 11E
Problem 12E
Problem 13E
Problem 14E
Problem 15E
Problem 16E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
Problem 23E
Problem 24E
Problem 25E
Problem 26E
Problem 27E
Problem 28E
Problem 29E
Problem 30E
Problem 31E
Problem 32E
Problem 34E
Problem 35E
Problem 36E
Problem 37E
Problem 38E
Problem 39E
Problem 40E
Problem 41E
Problem 42E
Problem 44E
Problem 45E
Problem 46E
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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

If a particle with mass m moves with position vector r(t), then its angular momentum is defined as L(t) = mr(t) × v(t) and its torque as τ(t) = mr(t) × a(t). Show that L′(t) = τ(t). Deduce that if τ(t) = 0 for all t, then L(t) is constant. (This is the law of conservation of angular momentum.)

To determine

To show: The L′(t)=τ(t) .

Explanation

Given data:

L(t)=mr(t)×v(t) (1)

τ(t)=mr(t)×a(t)

Formula used:

From differentiation rule,

ddt[u(t)×v(t)]=u′(t)×v(t)+u(t)×v′(t)

Take differentiation with respect to t on both sides of equation (1).

ddt[L(t)]=ddt[mr(t)×v(t)]=mddt[r(t)×v(t)]=m[r′(t)×v(t)+r(t)×v′(t)]=m[v(t)×v(t)+r(t)×a(t)] {∵r′(t)=v(t),v′(t)=a

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Chapter 13.4,
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