Constant-speed ramps

Oscar M. Perdomo

Vol. 275 (2015), No. 1, 1–18

DOI: 10.2140/pjm.2015.275.1

Abstract

It is easy to show that if the kinetic coefficient of friction between a block and a ramp is ${\mu }_k$ and this ramp is a straight line with slope $-{\mu }_k$, then this block will move along the ramp with constant speed. A natural question to ask is the following: besides straight lines, are there other shapes of ramps such that a block will go down the ramp with constant speed? Here we classify all possible shapes of these ramps, and, surprisingly, we show that the planar ramps can be parametrized in terms of elementary functions: trigonometric functions, exponential functions and their inverses. They provide basic examples of curves explicitly parametrized by arclength. A video explaining the main results in this paper can be found at http://youtu.be/iBrvbb0efVk.

Keywords

friction, constant speed, ramps, sliding

Mathematical Subject Classification 2010

Primary: 53Z05

Milestones

Received: 10 May 2013

Revised: 5 October 2013

Accepted: 17 October 2013

Published: 12 April 2015