A flat metal plate left outdoors obeys Newton's law of cooling,

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Ordinary differential equations

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

Problem

\dot{T} = -(T - T_a(t))/\tau

, where

T(t)

is the plate's temperature,

\tau = 2.0\,\mathrm{h}

is its (constant) time constant, and the air temperature varies over the day as

T_a(t) = 15.0^{\circ}\mathrm{C} + 5.0^{\circ}\mathrm{C}\cos(\omega t), \qquad \omega = \frac{\pi}{12}\,\mathrm{rad\,h^{-1}}

(

t

in hours, so the period is

24\,\mathrm{h}

). At

t = 0

the plate's temperature is

T(0) = 10.0^{\circ}\mathrm{C}

. Treat the plate as having a single uniform temperature and

\tau

as fixed (independent of

T

). (a) Write the equation in the standard linear form

\dot{T} + T/\tau = s(t)

and explain in one line why it cannot be solved by separating variables. (b) Using the integrating factor

\mu(t) = e^{t/\tau}

(or, equivalently, treating the oscillating part of

s(t)

as the real part of a complex exponential), show that the steady-state part of

T(t)

has the form

T_{\mathrm{ss}}(t) = 15.0^{\circ}\mathrm{C} + \frac{5.0^{\circ}\mathrm{C}}{\sqrt{1+\omega^2\tau^2}}\cos(\omega t - \varphi), \qquad \varphi = \arctan(\omega\tau),

and evaluate the amplitude and

\varphi

numerically. (c) Impose

T(0) = 10.0^{\circ}\mathrm{C}

to find the coefficient

C

of the transient term

Ce^{-t/\tau}

, so that

T(t) = T_{\mathrm{ss}}(t) + Ce^{-t/\tau}

. (d) Find the time after which the transient has fallen to below

1\%

of its value at

t=0

Related theory

Ordinary differential equations

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