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Newton’s Law of Cooling. Newton’s law of cooling states
Chapter 1, Problem 15E(choose chapter or problem)
Newton’s law of cooling states that the rate of change in the temperature \(T(t)\) of a body is proportional to the difference between the temperature of the medium \(M(t)\) and the temperature of the body. That is,
\(\frac{d T}{d t}=K[M(t)-T(t)]\),
where K is a constant. Let \(K=1\ (\min )^{-1}\) and the temperature of the medium be constant, \(M(t) \equiv 70^{\circ}\). If the body is initially at \(100^{\circ}\), use Euler’s method with \(h=0.1\) to approximate the temperature of the body after
(a) 1 minute.
(b) 2 minutes.
Equation Transcription:
Text Transcription:
T(t)
M(t)
dT over dt=K[M(t)-T(t)]
K=1 (min)^-1
M(t)70deg
100deg
h=0.1
Questions & Answers
QUESTION:
Newton’s law of cooling states that the rate of change in the temperature \(T(t)\) of a body is proportional to the difference between the temperature of the medium \(M(t)\) and the temperature of the body. That is,
\(\frac{d T}{d t}=K[M(t)-T(t)]\),
where K is a constant. Let \(K=1\ (\min )^{-1}\) and the temperature of the medium be constant, \(M(t) \equiv 70^{\circ}\). If the body is initially at \(100^{\circ}\), use Euler’s method with \(h=0.1\) to approximate the temperature of the body after
(a) 1 minute.
(b) 2 minutes.
Equation Transcription:
Text Transcription:
T(t)
M(t)
dT over dt=K[M(t)-T(t)]
K=1 (min)^-1
M(t)70deg
100deg
h=0.1
ANSWER:
Solution:-
Step1
Given that
We have to approximate the temperature of the body after
(a) 1 minute.
(b) 2 minutes
By using Euler’s method with h = 0.1.
Step2
We have
-------(1)
Where k is a constant and k=1
Medium be constant M(t)=
Let T(0) be initial body temperature
So, T(0)=
By using Euler’s method
[from (1)]
Step3
Put h=0.1, M(t)=70 and k=1 we get
=
=
So, =
and
Step4
(a) 1 minute.
For t=1 minute and
Value of x |
Time (t) in minutes |
Temperature in degrees= |
0 |
|
|
1 |