Forced harmonic motion of a particle subject to viscous drag

An electrically charged water droplet of mass m is acted on by an oscillating electric field so that the driving force on the droplet is F = F cos ( ?t). The viscous drag force on the particle caused by the air is ?u, where u is the velocity.

1. What is the differential equation for the velocity u(t) of the particle?

2. What is the corresponding equation for the complex velocity amplitude u( ?)?

3. Solve the equation for the complex velocity amplitude and determine the amplitude (magnitude) and phase angle of the resulting velocity as functions of frequency.

4. What are the amplitude and phase angle of the complex displacement amplitude?

5. Do the above without the use of complex amplitudes.

   SOLUTION

1. m (t) + ?u = F cos ( ?t)

2. ? i ?mu( ?) + ?u( ?) = F

3. ,

   where tan = ?m/ ?. (The phase angle of the denominator is arctan ( ? ?m/ ?) so that the phase angle of the inverse is = arctan ( ?m/ ?), making the phase angle of u( ?) equal to arctan ( ?m/ ?).

   At low frequencies, ?m << ?, the viscous force dominates and...