Particle sliding on a turntable in the presence of frictional forces

TL;DR

This paper analyzes the motion of a particle on a turntable with friction, deriving equations, finding approximate solutions, and identifying an escape speed that depends on initial conditions and position.

Contribution

It introduces a detailed mathematical model for particle motion on a turntable with friction, including approximate solutions and methods to measure friction coefficient.

Findings

Defined an escape speed depending on position and initial velocity.

Derived equations of motion with frictional forces.

Suggested a method to measure the coefficient of friction.

Abstract

Motion of a point particle sliding on a turntable is studied. The equations of motion are derived assuming that the table exerts frictional force on the particle, which is of constant magnitude and directed opposite to the direction of motion of the particle relative to the turntable. After expressing the equations in terms of dimensionless variables, some of the general properties of the solutions are discussed. Approximate analytic solutions are found for the cases in which (i) the particle is released from rest with respect to the lab frame and, (ii) the particle is released from rest with respect to the turntable. The equations are then solved numerically to get a more complete understanding of the motion. It is found that one can define an escape speed for the particle which is the minimum speed required to get the particle to move off to infinity. The escape speed is a function of the distance from the center of the turntable and for a given distance from the center, it depends on the direction of initial velocity. A qualitative explanation of this behavior is given in terms of the fictitious forces. Numerical study also indicates an alternative way for measuring the coefficient of friction between the particle and the turntable.