Newton's law of cooling states that- = -k (x- A), where x(t) is the temperature of an object at dt time t, A is the temperature of the object's surroundings and k is a constant. a) If the ambient temperature A is constant, only one choice of sign for k makes physical sense - either k is positive or k is negative. Explain which choice is correct. b) Suppose the object sits outside, in Potsdam, over the course of a year. The ambient temperature will vary with time - suppose A = Ao cos(wt), where Ao and wo are constants. i. What might be reasonable values for A, and wo? 0) the object is at room temperature (so x(0) = T, ii. Let's say that initially (at time t = where To represents room temperature). %3D What impact, in the long-run, does the value T, have on the temperature x(t) of the obiect?
Newton's law of cooling states that- = -k (x- A), where x(t) is the temperature of an object at dt time t, A is the temperature of the object's surroundings and k is a constant. a) If the ambient temperature A is constant, only one choice of sign for k makes physical sense - either k is positive or k is negative. Explain which choice is correct. b) Suppose the object sits outside, in Potsdam, over the course of a year. The ambient temperature will vary with time - suppose A = Ao cos(wt), where Ao and wo are constants. i. What might be reasonable values for A, and wo? 0) the object is at room temperature (so x(0) = T, ii. Let's say that initially (at time t = where To represents room temperature). %3D What impact, in the long-run, does the value T, have on the temperature x(t) of the obiect?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Linear example:
Problem 3 (modified from exercise 1.4.10):
dx
Newton's law of cooling states that
dt
= -k (x - A), where x(t) is the temperature of an object at
time t, A is the temperature of the object's surroundings and k is a constant.
a) If the ambient temperature A is constant, only one choice of sign for k makes physical sense -
either k is positive or k is negative. Explain which choice is correct.
b) Suppose the object sits outside, in Potsdam, over the course of a year. The ambient temperature
will vary with time suppose A
Ao cos(wt), where Ao and wo are constants.
%3D
i. What might be reasonable values for A, and w,?
ii. Let's say that initially (at time t = 0) the object is at room temperature (so x(0) = To,
where T, represents room temperature).
What impact, in the long-run, does the value T, have on the temperature x(t) of the
object?
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