Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 16 - Section 16.2 - Line Integrals - 16.2 Exercise - Page 1086: 47

Answer

a) It has been proved that a constant force field does zero work on a particle that moves once uniformly around the circle $x^2 +y^2=1$ b) Yes

Work Step by Step

a) Let us consider $F(x,y)=A(x,y) i+B(x,y) j$ Work done, $W=\int_C F\cdot ds=\int_C a dx +\int_C b dy$ or, $W=\int_0^{2 \pi} a [-\sin t dt]+\int_0^{2 \pi} b [\cos t dt]=a(0)+b(0)=0$ Hence, it has been shown that a constant force field does zero work on a particle that moves once uniformly around the circle $x^2 +y^2=1$ b) Let us consider $F(x,y)=A(x,y) i+B(x,y) j$ Work done, $W=\int_C F\cdot ds=\int_C A dx +\int_C B dy=\int_0^{2 \pi} [k\cos t dt][-\sin t] dt+\int_0^{2 \pi} k \sin t$ and $W=k \int_0^{2 \pi} -\cos t \sin t +\sin t \cos t dt=0$ Yes, for a force field $F(x)=kx$
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