Observe that in case you’re studying this on a cellular machine, a number of the equations don’t look proper. Specifically, one factor I seen is that the sq. root radical over a fraction seems to solely be over the numerator. All such sq. roots are over the denominator too. The equations look proper (to me) when considered on a non-mobile machine (e.g., my laptop computer).
This put up is a couple of subject in physics, which I think is probably not of curiosity to many common TIE readers. However, it may curiosity irregular (?) readers.
Why physics? It was my undergrad main, and I’m now serving to my daughter by means of AP Physics. I’m discovering myself taking place some rabbit holes, exploring issues I didn’t study in school. Some aren’t in any textbook I’ve at my fingertips or on-line (as far as I can inform). This put up is about one such factor. I used AI to examine the mathematics and logic. In case you see an issue, please let me know.
What This Put up Exhibits
Our place to begin is a widely known property of a bodily pendulum (additionally referred to as a compound pendulum), the invention of which is credited to Christian Huygens: you possibly can pivot the pendulum from two totally different factors and get the identical interval (they’re isochronal). Given one pivot level, Huygens and every other primary physics textual content that covers this subject, exhibits methods to discover “the” different isochronal pivot level.
What this put up exhibits is that, the truth is, there are an infinite variety of isochronal factors (therefore the quotes round “the” within the earlier sentence). Furthermore, all of them lie on two circles concerning the heart of mass (CM).
The First Isochronal Circle
The derivation for the interval of a bodily pendulum will be discovered on-line or in lots of textbooks. The interval T about pivot level P is given by
the place I is the second of inertia concerning the pivot level, m is the pendulum’s mass, g is the acceleration as a consequence of gravity, and d is the gap between P and the CM. From the parallel axis theorem
the place the primary time period is the second of inertia concerning the heart of mass. Subsequently,
The Second Isochronal Circle
As mentioned in lots of locations however not proven in a easy manner (that I’ve seen), there’s a pivot level isochronal with P and never on the circle described above. It’s referred to as the heart of oscillation and we’ll label it level Q. The tip of this put up has an easy proof that the isochronal level Q is a distance L from P by means of the CM the place
Outline d’ = L – d as the gap between Q and the CM. The bodily pendulum has a second of inertia I’ about level Q. We are able to plug d’ and I’ into the equations supplied within the part above (The First Isochronal Circle). The conclusion follows by the identical argument as above that the interval is a continuing T for any level a distance d’ from the CM. We all know it’s T (the identical interval as within the earlier part) as a result of P and Q are isochronal pivot factors. Thus there are an infinite isochronal pivot factors that lie on a circle of radius d’ centered on the CM. That is the second isochronal circle.
Or, to sum up, Huygens and numerous others open our eyes to the truth that there are two isochronal factors, P and Q. What can be true is that there are two isochronal circles concerning the CM, one accommodates P, the opposite Q.
That is according to the apparent undeniable fact that the interval of oscillation of a uniform density rod pivoting at one finish is identical as that pivoting on the opposite finish (an isochronal circle goes by means of each ends). Or, the interval of oscillation is identical for any level of a round disk of uniform density equidistant from the middle (isochronal circles are concentric concerning the CM). These apparent information are clear to us from symmetry.
What’s given above exhibits that the round isochronal symmetry is there even for uneven (arbitrarily formed, with non-uniform density) bodily pendulums. This isn’t intuitive (to not me anyway). But, the one factor that isn’t fastened for all factors regarding a given pendulum within the expression for T is the gap of the pivot level from the CM. With that, isochronal circularity is unavoidable.
Conclusion
Discover that d will be any optimistic distance, giving rise to any optimistic interval T. So, by the above arguments, all isochronal pivot factors lie on pairs of concentric circles centered concerning the CM. (There’s a small technicality that for some (OK, an infinite variety of) values of d, some isochronal factors is probably not inside the physique of the pendulum. That doesn’t imply they’re not isochronal with the opposite factors on the circle(s) related to d (or d’). It simply signifies that truly getting the pendulum to pivot round such some extent is difficult in follow. It’d take a massless, unbending help from that time to the pendulum’s CM.)
Proof That the Heart of Oscillation is Isochronal (Or Pivot Q Has the Identical Interval as Pivot P)
Above, I promised this proof. Skip it in case you’re prepared to belief Huygens and numerous different physics texts, a few of which state this with out proof.
With all of the phrases as outlined within the put up, begin with some preliminary stuff to get an expression of d’ when it comes to d that might be helpful later. Recall that
By the parallel axis theorem, and as famous within the put up
Subsequently,
And, simplifying,
Utilizing this and the definition of d’ (= L – d) from the put up, we’ve
In order that
Plug within the expression for d’ from above. (Because of this we did that preliminary work.)
There’s tons to cancel to simplify. Go forward and do this on a scrap of paper or in your head. You’ll get