   Chapter 13.4, Problem 44E

Chapter
Section
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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