A bead is carried up a spiral wire ,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Mathematical Methods for Physics, Ch Line, surface and volume integrals

Problem

A bead is carried up a spiral wire r(t)=(cos⁡t, sin⁡t, t) m\mathbf{r}(t) = (\cos t,\ \sin t,\ t)\ \mathrm{m}, t∈[0,2π]t\in[0,2\pi], from (1,0,0) m(1,0,0)\ \mathrm{m} to (1,0,2π) m(1,0,2\pi)\ \mathrm{m}, one full turn round the zz axis. Two force fields act on it in turn, with xx, yy, zz in metres and the forces in newtons: F=(2xy+z2, x2, 2xz),G=(2xy+z2−y, x2+x, 2xz).\mathbf{F} = (2xy+z^2,\ x^2,\ 2xz), \qquad \mathbf{G} = (2xy+z^2-y,\ x^2+x,\ 2xz). ((a)) Find the work done by F\mathbf{F} on the bead along the spiral. ((b)) Find the work done by G\mathbf{G} along the same spiral. ((c)) Find the work done by each field along the straight vertical route from (1,0,0) m(1,0,0)\ \mathrm{m} to (1,0,2π) m(1,0,2\pi)\ \mathrm{m}, and explain why one field gives the same answer on both routes and the other does not.

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