A small steel ball falls from rest through a viscous oil

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Kinematics and Newtonian dynamics of particles

Mathematical Methods for Physics, Ch First-order differential equations and physical modelling

Problem

A small steel ball falls from rest through a viscous oil. Its speed is modelled by linear drag,

\dot{v} = g - v/\tau

, with time constant

\tau = 8.0\,\mathrm{ms}

(this linear law is a modeller's choice, not a law of physics). Neglect buoyancy. ((a)) Without solving the equation, use its direction field to state the equilibrium speed

v^{*}

and, from the sign of

\partial\dot{v}/\partial v

v^{*}

, say whether it is stable or unstable. ((b)) State the fraction of

v^{*}

reached after

\tau

2\tau

and

3\tau

, to three significant figures.

Related theory

Ordinary differential equations

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