Mathcad Community Challenge July 2026: Tautochrone and Isochronous Curves

This month’s challenge is about tautochrone curves and isochronous curves. They are very similar. Both involve placing an object on the curve and allowing it to slide without friction. With a tautochrone, an object placed at any point on the curve will take the same time to slide to the bottom. With isochronous curves, objects allowed to slide from any part of the curve will have the same period of oscillation. You can think of the tautochrone as “half” of the isochronous curve, as it is only concerned with how long it takes the object to reach the bottom.

In this scenario, the “top” of the tautochrone curve is at a height of 10 meters on the left side. The end point is at ( 0, 0 ). (The “top” of the isochronous curve on the other side is 10 meters on the right side.) Design a curve such that an object placed at any point on the curve will take the same time to slide without friction to the end point under the effect of uniform gravity.

Your challenge, should you choose to accept it, is to solve any of the following in Mathcad Prime:

Challenge 1: Derive the equation of the tautochrone or isochronous curve.

Challenge 2: Graph the curve on an XY Plot or in the Chart Component. (Note that the solutions to this problem are easy to find online. You do not need to complete Challenge 1 to perform this step.)

Challenge 3: Find a function for the velocity of the object at the bottom based on either the starting height of the object, the starting horizontal distance, or the distance along the curve. (Hint: one of these is easy.)

Challenge 4: Graph the results of Challenge 3.

Challenge 5: Calculate the period of oscillation in this scenario for the isochronous curve.

Challenge 6: How does the equation for the tautochrone / isochronous curve compare to the brachistochrone curve?

Have fun! As always, post any question regarding the parameters of this challenge. (No friction, uniform gravity at the Earth’s surface.)

Find the Mathcad Community Challenge Index and Guidelines here!