Mathematics

Mathematics

Problem

A wheel of radius 0.6 m0.6\,\mathrm{m} turns through an angle of 72∘72^{\circ}. (a) Find the distance travelled by a point on the rim of the wheel. (b) Determine the area swept out by a spoke of the wheel during this rotation. (c) If the wheel completes 5050 such rotations, calculate the total distance covered by the point on the rim.
Intuitive explanation

The key is converting between angular measure and linear/area quantities using the radius as your bridge. When a wheel rotates through an angle θ\theta (in radians), a point on the rim traces an arc whose length is s=rθs = r\theta, and a spoke sweeps out a sector whose area is A=12r2θA = \frac{1}{2}r^2\theta. Start by converting degrees to radians—that's where most errors happen—then apply these formulas directly, and scale up for multiple rotations.

Intuitive explanation

The key is converting between angular measure and linear/area quantities using the radius as your bridge. When a wheel rotates through an angle θ\theta (in radians), a point on the rim traces an arc whose length is s=rθs = r\theta, and a spoke sweeps out a sector whose area is A=12r2θA = \frac{1}{2}r^2\theta. Start by converting degrees to radians—that's where most errors happen—then apply these formulas directly, and scale up for multiple rotations.

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