From Equations of Motions To Mathematical Identities
From Equations of Motions To Mathematical Identities
From Equations of Motions To Mathematical Identities
From Mechanical Damping to Mathematical Identities -Abhijit Kar Gupta We have encountered the following mathematical identity: ( ) , the proof
of this is done by binomial expansion and then applying the limits. Can we arrive at this through some other ways; can we think how physics can be of help in deriving such a limiting identity, in particular? Normally, we employ mathematical identities or expressions in order to form ideas about physical systems. Here we try to build an insight approaching from the other side. Consider the motion of a particle in one direction under a damping where the damping force is proportional to the square of the speed:
Here, is the damping constant and we consider the mass of the particle is unity for no loss of generality. Now to solve this,
If we consider, at
For , in the limit of no damping, no deceleration, we have with initial speed. Thus in the limit of , ( If , ( If we consider, ) , ) ) )
(2)
The proof looks interesting! However, a little thinking tells us that the expression (2) is actually ( ) , in the limit of . This relation is nothing but the first two terms in the Taylor Series expression of exponential: ( ) ( ) .
We can proceed further. Consider the distance traveled by the particle in time : , where the deceleration is time, we can write, have, . Thus we have, ( ( Remember, the above is true for For , we write, ( ( ) ) ) limit. for ) . In the limit of small damping and for small . If we use this in the expression (1), we ( ( ) )
If we think the above exercise is a kind of fun where we ultimately see an intimate connection of mathematics and physics, we can do similar exercises and gain some more insight. For example, let us now consider the damping force to be proportional to speed, which is usually true in most circumstances.
We find through a similar analysis as we did previously, ( Again for (in the limit of zero damping), ( So for , and for , ) ), where
What if we consider a particle falling through a viscous fluid? Here we have to solve the following equation: [Consider
at
where we have considered initial position at The above expression is rewritten in the form:
This is again the first two terms of the Taylor series for exponential term. Also, as we consider the initial speed to be zero, Thus [ ( )] , the particle is falling under gravity.
Therefore,
The expression on right is the first three terms of the Taylor series expansion: ( ) ( ) .
Let us consider another interesting byproduct of such an analysis, which gave us back some well-known mathematical identities and relations. Can we ask whether a damping force proportional to speed cube is ever possible? Does Nature support this? At least, can we shed some light if this is possible mathematically? Imagine,
Integrating we find,
, where at (
Next, consider a similar analysis as before, ( ) After a little manipulation we see, ( have . Therefore, we see such a damping force ( . Thus we will have to conclude, ) will never exist! as we )
Conclusion:
All we have done is revisiting the well-known equations of motion where damping forces are velocity dependent. In the process of solving we added little twists at the end which led us to well-known mathematical identities in some limit. Also, we may have gained a different insight how Nature does not violet our mathematical constructs. The moral of the story is that a greater understanding of how physics and mathematics are interconnected can be extracted from playing with the simple equations. Postscript: The treatment and idea of this simple work is generated after reading a beautiful popular science book, Mrs. Perkinss Electric Quilt by Paul J. Nahin (Pub: Princeton University Press).
Any comment and criticism by teachers, students and enthusiasts would be most welcome. Dr. Abhijit Kar Gupta Department of Physics, Panskura Banamali College Panskura, East Midnapore, PIN 721152, WB, India e-mail: kg.abhi@gmail.com