In the previous section, we saw an example which demonstrated the basics of mathematical modeling. In this section, we will see another example.
Example 2:
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The situation is:
A physicist (scientist who specializes in the field of physics) wants to understand the motion of simple pendulum. A simple pendulum consists of a mass (called bob) attached to one end of a string. The other end of the string is fixed at a point.
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This situation can be converted into a mathematical problem. For that,
we engage in the process of mathematical modeling. The final result can
be obtained in 4 steps:
Step 1:
The physicist
studies the situation. He realizes that, the factors that may be affecting the motion of a pendulum are:
(i) Period of oscillation (T)
(ii) Mass of the bob (m)
(iii) Effective length of the pendulum (l). This is the distance between the point of suspension to the center of mass of the bob.
(iv) Acceleration due to gravity (g)
• Now, the physicist tries to confirm that, the above four are indeed the parameters. For that, he performs simple experiments.
• Time period of the following two pendulums are determined experimentally:
(i) Pendulum with effective length l and mass of bob m1.
(ii) Pendulum with same effective length l but mass of bob m2.
• He finds that, there is no appreciable change in T.
• Again, T of following two pendulums are determined experimentally:
(i) Pendulum with effective length l1 and mass of bob m.
(ii)Pendulum with a different effective length l2 but same mass m.
• He finds that, there is appreciable change in T.
• So he concludes that:
♦ m is not an essential parameter.
♦ l is an essential parameter.
•
We can write:
Step 1 involves studying the situation and identifying the parameters. Essential parameters should be carefully identified.
Step 2:
This can be written in order:
(i) The physicist repeats the experiment with same m and different l. He notes T in each trial.
(ii) Using the observations, he plots a graph with T along the y-axis and l along the x-axis.
• The graph thus obtained is a parabola.
(iii) Equation of a parabola is of the form y2 = kx.
• So he can conclude that, the relation between T and L will be of the form T2 = kl
(iv) From the values of T and l of the various trials, k can be obtained. He gets:
$k = \frac{4 \pi^2}{g}$
• Substituting in (iii), we get:
$\begin{array}{ll}{} &{T^2} & {~=~} &{\frac{4 \pi^2}{g} l} &{} \\
{\Rightarrow} &{T} & {~=~} &{\sqrt{\frac{4 \pi^2}{g} l}} &{} \\
{\Rightarrow} &{T} & {~=~} &{2 \pi \sqrt{\frac{l}{g}}} &{} \\
\end{array}$
•
We can write:
Step 2 involves drawing the necessary diagrams and writing the relevant mathematical equations. In short, we write a mathematical problem
in this step. Any mathematical problem will have a definite solution.
So in this step, it is necessary to recheck all the works done thus far.
Step 3:
This can be written in order:
(i) The physicist puts the equation into actual use.
(ii) He assumes two values for l. They are 225 cm and 275 cm.
(iii) When l = 225 cm, he gets:
$\begin{array}{ll}{} &{T} & {~=~} &{2 \pi \sqrt{\frac{l}{g}}} &{} \\
{} &{} & {~=~} &{2 \pi \sqrt{\frac{2.25}{9.81}}} &{} \\
{} &{} & {~=~} &{3.04 ~\text{sec}} &{} \\
\end{array}$
(iv) When l = 275 cm, he gets:
$\begin{array}{ll}{} &{T} & {~=~} &{2 \pi \sqrt{\frac{l}{g}}} &{} \\
{} &{} & {~=~} &{2 \pi \sqrt{\frac{2.75}{9.81}}} &{} \\
{} &{} & {~=~} &{3.36 ~\text{sec}} &{} \\
\end{array}$
•
We can write:
Step 3 involves the actual application of the result obtained in step 2. Calculators or digital computers can be used for
lengthy problems.
Step 4:
This can be written in order:
(i) The physicist tries to confirm the results obtained in step 3. For that, he performs a number of trials of the actual experiment.
(ii) When mass (m) of the bob is 385 gms and l is 275 cm, T is 3.371 sec
• This T obtained experimentally, is comparable with the 3.36 sec that he got by the mathematical calculations in step 3.
(iii) When m is 385 gms and l is 225 cm, T is 3.056 sec
• This T obtained experimentally, is comparable with the 3.04 sec that he got by the mathematical calculations in step 3.
(iv) When m is 230 gms and l is 275 cm, T is 3.351 sec
• This T obtained experimentally, is comparable with the 3.36 sec that he got by the mathematical calculations in step 3.
(v) When m is 230 gms and l is 225 cm, T is 3.042 sec
• This T obtained experimentally, is comparable with the 3.04 sec that he got by the mathematical calculations in step 3.
(vi) However, there are some small errors. For example, in (ii) above, we can find an error of (3.371 - 3.36) = 0.011.
• But such errors are small. So the mathematical model in step 3 is acceptable.
• We can conclude that:
♦ T is directly proportional to l.
♦ T is inversely proportional to g.
(vii) We see that, the mathematical model is in good agreement with the practical values. But small errors are also seen.
• These errors are due to:
♦ mass of the string, which we did not consider.
♦ resistance of the air, which also we did not consider.
• When the physicist sees the errors, he investigates further. Such an investigation will help to find the reasons for the errors.
• Once the reasons are found out, the mathematical model can be improved. The improved model may contain many complex equations.
• We can surely say that, a simple mathematical model is a good starting point.
(viii) We can write:
• In step 4, we do validation and interpretation.
◼ What is validation?
The answer can be written in 4 steps:
(a) We have two items:
♦ Results obtained from the mathematical model.
♦ Known facts about the real problem.
(b) We compare the two items
(c) If there is no appreciable difference, the mathematical model can be considered to be valid.
(d) If there is appreciable difference, it cannot be considered to be valid.
◼ Suppose that a mathematical model is found to be valid. Once found valid, we have to do the interpretation. What is interpretation?
The answer can be written in 2 steps:
(a) We have the results obtained from the mathematical model. We state, why and how those results are obtained. In our present case, the physicist state that, T is directly proportional to l and inversely proportional to g.
(b) We also state why and how some errors (if any) occur. In our present case, the physicist state that, mass of the string and air resistance are the cause of the errors.
So we have seen the mathematical model related to the oscillation of a simple pendulum. In the next section, we will see a mathematical model related to inequalities.
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