Cycloids: Parametric Equations, Calculus, and Motion

All flashcards

Cycloid

The path traced by a fixed point on a circle rolling without slipping along a straight line.

Cycloidal cusp

The sharp point where the tracing point touches the line at the end of an arch.

Tautochrone

A curve on which ideal gravity-driven travel to the lowest point takes equal time from different starting positions.

Involute

The path traced by the free end of a taut string as it unwinds from a curve.

Cycloid parametric equations

For a generating circle of radius $r$, the standard equations are $x=r(t-\sin t)$ and $y=r(1-\cos t)$.

Quiz questions

1. For a circle of radius $r$ rolling to the right without slipping, with $t=0$ at a cusp, which equations give the traced point's position?Answer: $x=r(t-\sin t),\quad y=r(1-\cos t)$
2. Under uniform gravity, what does the brachistochrone property of an appropriately oriented cycloid describe?Answer: A path of fastest descent between suitable points.
3. For an ideal cycloidal pendulum with length $L=4r$, which angular-frequency relation applies?Answer: $\omega^2=\frac{g}{L}=\frac{g}{4r}$
4. What path is called the involute of a curve?Answer: The path of a taut string's free end as the string unwinds from the curve.
5. What trajectory can a charged particle initially at rest follow in uniform, perpendicular electric and magnetic fields?Answer: A cycloidal trajectory.