Category:Cycloid has Tautochrone Property

This category contains pages concerning Cycloid has Tautochrone Property:

Let a wire $AB$ be curved into the shape of an arc of a cycloid such that:

$A$ is at the cusp

$B$ is the highest point of the arc

and inverted so that its cusps are uppermost and on the same horizontal line.

Thus $B$ is the lowest point of the arc.

Let $AB$ be embedded in a constant and uniform gravitational field where Acceleration Due to Gravity is $g$.

Let a bead $P$ be released from anywhere on the wire between $A$ and $B$ to slide down without friction to $B$.

Then the time taken for $P$ to slide to $B$ is:

$T = \pi \sqrt {\dfrac a g}$

independently of the point from which $P$ is released.

That is, a cycloid is a tautochrone.