A bead threads a stiff spiral wire wound round a vertical pole

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 threads a stiff spiral wire wound round a vertical pole. As the bead goes round, its horizontal position traces a circle of radius 0.3 m0.3\ \mathrm{m} centred on the pole; for every complete turn (an angle of 2π rad2\pi\ \mathrm{rad}) the bead rises a further 0.5 m0.5\ \mathrm{m} up the pole. Take the angle turned, θ\theta, measured from the bead's starting position, as the parameter. Place the origin on the pole's axis at the height of the bead's starting point, with the bead starting on the positive xx axis. ((a)) Write the bead's position as a parameterised curve r(θ)\mathbf{r}(\theta). ((b)) Find the length of wire the bead has passed over after three complete turns.

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