Question

A particles moves over the semicircle C : r(I)-costit snIJ, (0 to the force F(x,y)-e i +xe' j. Find the work done. z) while subject t

Answer 1

Partial moves over semicircle c: r(t) = cos t i^^+ sin t j^^0 lessthanorequalto t lessthanorequalto pi Force, f(x, y) = e^y i^^+ xe^y j^^x = cos t, y = sin t F(x(t), y(t)) = e^sin t i^^+ cos t middot e^sin t j^^Work done = integral_c F middot dr = integral_pi_0 F middot dr/dt middot dt [dr/dt = -sin t i^^+ cos t j^^] = integral^pi_0 < e^sin t, cos t e^sin t > middot < -sin t, cos t > dt = integral^pi_0 (-sin t e^sin t + cos^2 t e^sin t) dt = integral^pi_0 (-sin t + cos^2 t) e^sin t dt evaluating Indefinite integral, I = integral(-sin t + cos^2 t)e^sin t dt Let z = sin t implies dz = cos t dt implies dt = 1/squareroot 1 - z^2 dz I = integral (-z + (1 - z^2))e^z 1/squareroot 1 - z^2 dz = integral(-z/squareroot 1 - z^2 + squareroot 1 - z^2) e^z dz