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HINT: for finding out the turning/angular acceleration say $\alpha$, you need to know the angular displacement say $\theta$ as a function of the time $t$ then $$\alpha=\frac{d^2\theta}{dt^2}$$


This is an excellent question. It is pedantic to whine about non differentiability ,however there is in fact a point to be made on that topic . We seem to be conflating the derivative dy/dx with the time derivative dx/dt(x is the position here). One may have a differentiable path but still the instantaneous velocity can remain undefined. Whether the converse ...


No, it's not possible, because one of the underlying assumptions of kinematics is that all paths are at least twice differentiable. Before you complain about this requirement, remember that physics is about building models that can be used to describe and predict measurements. Measurements always have some amount of uncertainty, and even if you suppose that ...


The length along any segment of the Koch snowflake is infinite. It has finite area but infinite perimeter. So, for a particle to move from one place on the snowflake to another it would have to travel an infinite distance. This is why differentiability is important.


There isn't a way to derive $\pi$ because it's a fundamental constant and not something that can be derived. However there are lots and lots of ways to approximate $\pi$. I believe that Archimedes was the first person to write down such an approximation (in 250BC), and we've been developing better and better ways of approximating $\pi$ since. One of the ...


The value of $\pi$ cannot be derived from QM. The article discusses the observation that $\pi$ (in fact the Wallis product for $\pi$) appears in a derivation of the energy levels of hydrogen. I don't personally find that remarkable for math or physics or particularly significant.


Conformal maps tend to be the exception rather than the rule. In general, if a transformation $T:S\to S$ on some $n$-dimensional space $S$ is of interest in physics, it will not be conformal. (Indeed, there's generally no guarantee of a useful notion of angle in that space, but even if there is such a guarantee then $T$ is still not likely to be conformal.) ...


If we have a function $f$ that is a function of several variables $p$, $q$, $r$, etc then we can write a total derivative of $f$ as: $$ df = \frac{\partial f}{\partial p} \text{d}p + \frac{\partial f}{\partial q} \text{d}q + \frac{\partial f}{\partial r} \text{d}r + \, ... $$ If we're holding all the variables constant except for $p$, so $\text{d}q = ...


It might not be as mysterious as it appears. Pi is only the consequence of converting from one coordinate system (orthogonal) to another (spherical). This exceeds my cleverness, but I think if one created another coordinate system (with some axes 30 degrees apart instead of 90) then there would be some square roots (which are also irrational) in the ...


Strange question. This should probably be on MathSE. There are a million proofs of the irrationality of pi, and why it has to be irrational. You need some reasonable Mathematical knowledge to understand them - it's not like proving the irrationality of root 2. I encourage you to have a look!! As for the whole "it just is" thing... "It just is" isn't the ...


Here is a fun paper on using Möbius inverse formula. Nan-xian Chen, Modified Möbius inverse formula and its applications in physics, Phys. Rev. Lett. 64, 1193 (1990).

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