This problem uses a partial-fractions integral and a complex-impedance check

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Electric current and circuits

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

Problem

This problem uses a partial-fractions integral and a complex-impedance check, both introduced in earlier chapters. ((a)) A steel ball bearing is released from rest into a tall column of glycerine. Its quadratic drag model gives

\dot{v} = g(1 - v^2/v_T^2)

, with terminal speed

v_T = 2.0\,\mathrm{m\,s^{-1}}

; neglect buoyancy. Separate the variables and integrate by partial fractions (the method of Chapter 3) to find the time

t_1

at which the ball reaches

90\%

v_T

, starting from

v(0)=0

. ((b)) An RC circuit with

R = 4.7\,\mathrm{k\Omega}

and

C = 22\,\mathrm{\mu F}

is driven from rest,

q(0)=0

, by a source

V(t) = 9.0\cos(150\,\mathrm{rad\,s^{-1}}\,t)\,\mathrm{V}

. Using the integrating factor for

\dot{q} + q/RC = V(t)/R

, write down the amplitude

q_0

and phase lag

\delta

of the steady-state charge,

q_{\text{steady}}(t) = q_0\cos(\omega t - \delta)

. Then check

q_0

and

\delta

independently using the complex impedance

Z = R + i/(\omega C)

of Chapter 6, with its time dependence

e^{-i\omega t}

Related theory

Ordinary differential equations

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