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68 votes
Accepted

Why aren't the lengths of the bars on a toy glockenspiel proportional to the wavelengths?

The answer to this question has significant overlap with my answer on piano tuning. There, I discuss how a thick wire has an extra restoring force, in addition to its tension, from its resistance to ...
knzhou's user avatar
  • 103k
49 votes

Why is the stress on a body not a vector?

Draw a square on an elastomer strip and stretch it: "OK, I get this:" The lengthwise load (comprising two force vectors, to the left and to the right) applies a stress state on the ...
Chemomechanics's user avatar
34 votes

At what speed does information move through the atoms of a rigid object?

A mechanical impulse will travel along an iron bar at the speed of sound (which will depend on the material and also possibly on the frequency and amplitude of the impulse). But in the video the ...
gandalf61's user avatar
  • 56.4k
28 votes

Why do we bend a book to keep it straight?

You have essentially discovered principles behind bending moments and structural engineering. As another poster stated, physically the structure you made is stronger, because to bend something (for ...
James's user avatar
  • 1,712
24 votes
Accepted

Why does the curve of a hanging chain not minimize the area below it?

What is wrong with your argument is this paragraph: If we imagine the chain as having many small segments, then the potential energy of each segment is $E_p=mgh$. As the number of small segments ...
John Hunter's user avatar
  • 13.7k
23 votes
Accepted

How can I adapt classical continuum mechanics equations in order to agree with general relativity?

The answer you're looking for seems to be contained in Rezzolla & Zanotti: Relativistic Hydrodynamics (Oxford U.P. 2013) https://books.google.com/books/?id=KU2oAAAAQBAJ but it is not a trivial ...
pglpm's user avatar
  • 3,753
23 votes

Why does the curve of a hanging chain not minimize the area below it?

To put the accepted answer in mathematical terms, if you have a curve $y(x)$, hanging fixed at $x_0$ and $x_L$ at an height $h=y(x_0)=y(x_L)$, of total length $L$ and mass $M$ then then linear mass ...
JalfredP's user avatar
  • 4,799
22 votes

Why aren't the lengths of the bars on a toy glockenspiel proportional to the wavelengths?

As knzhou identifies, the key difference between vibrations of a free beam and a string is that the restoring force is now provided by bending moments (proportional to $\frac{d^4y}{dx^4}$) rather than ...
sammy gerbil's user avatar
  • 27.4k
21 votes

Why do we bend a book to keep it straight?

When you bend a piece of material, the resistance is provided by stretching the material on the outside part of the bend, and compressing the material on the inside of the bend. A thin flat sheet ...
Daddyo's user avatar
  • 885
20 votes

Shape of a rotating rope with one free-end

Before developing the theory, I decided to first make an experiment in order to understand, what we are dealing with. A cylinder with a diameter of 11.5 cm is mounted on the motor shaft (I used an old ...
Alex Trounev's user avatar
  • 3,774
18 votes
Accepted

Why is the stress on a body not a vector?

We can put a wire under tensile stress by pulling each end with a force of equal magnitude. If the wire has an East-West alignment we need to pull its eastern end to the East and its western end to ...
Philip Wood's user avatar
16 votes
Accepted

How can transverse waves on a string carry longitudinal momentum?

A fake derivation We can rather easily compute a horizontal velocity for the string fi we assume that the total velocity vector is everywhere normal to the string (this assumption is not always valid,...
ACuriousMind's user avatar
  • 126k
14 votes
Accepted

Navier-Stokes Derivation

The traditional derivation of the Navier-Stokes equations starts by looking at a fluid parcel and the different fluxes over the surface in the integral form. The integral form is preferred as it is ...
2b-t's user avatar
  • 1,809
14 votes

At what speed does information move through the atoms of a rigid object?

The question in the title is formally contradictory. A model is always a simplification of the real world for the sake of understanding and making predicions (usually mathematically) A "rigid ...
James K's user avatar
  • 695
12 votes

Classical Field Theory - Continuum limit in forming the Lagrangian density and the elasticity modulus

(This explanation is adapted from Nicholas Wheeler Notes, nevertheless is self-contained, also a slightly modified version is published on my website A Sudden Burst of Physics, Math and more ): I'll ...
Keith's user avatar
  • 738
11 votes

Why do we bend a book to keep it straight?

The other answers so far are technically correct, but none of them really seem to give a commonsense/intuitive and simple answer. So I'll have a go at one. Imagine very slightly bending some kind of ...
Stilez's user avatar
  • 4,231
11 votes
Accepted

Is it possible to derive Navier-Stokes equations of fluid mechanics from the Standard Model?

One way to derive fluid dynamics is to start from the equations of motion for $N$ particles, and use these to compute the evolution of average quantities (like the density) of the distribution of ...
Andrew's user avatar
  • 51.1k
10 votes

How can transverse waves on a string carry longitudinal momentum?

You are absolutely right in everything you said. The momentum is non zero only if the wave has a longitudinal mode, which is in fact the realistic case. Moreover when this is the case, the wave ...
Diracology's user avatar
  • 17.8k
10 votes

What physical state of a wound string corresponds to a 'tuned' string?

I am addressing this part of the question why companies (such as the one above) sell distinct strings when the same notes could be achieved with the same string tuned to different tensions. Consider ...
march's user avatar
  • 8,199
10 votes
Accepted

Transverse waves in a rope: Why does tension not increase?

The short answer is that the elasticity does affect the wave speed. However, when people typically talk about the wave speed on a taut string they are referring to very small disturbances. In the ...
Michael M's user avatar
  • 1,950
10 votes
Accepted

What is the the Implication of Navier-Stokes Millenium Problem to Practice?

I've never tried: but does it really mean we have not one known solutions for $u(x,t)$ and a given $p(x,t)$ and $𝞶>0$? Not even some trivial ones? Hard to believe... can someone shed some light on ...
Kyle Kanos's user avatar
  • 28.4k
9 votes

Why do tall buildings have low resonant frequencies?

We can model a short building as a uniform cuboid of density $\rho$ occupying the region $$0 \le x \le L_x$$ $$0 \le y \le L_y$$ $$0 \le z \le L_z$$ with its mass given by $$M = \rho V = \rho A L_z = ...
Bio's user avatar
  • 843
9 votes

How can transverse waves on a string carry longitudinal momentum?

I) There are already several good answers. OP is asking about the momentum of the non-relativistic string with only transverse displacements, whose Lagrangian density usually is given as $$ {\cal L}_T ...
Qmechanic's user avatar
  • 206k
9 votes
Accepted

What is the difference between "Elastic limit" and "Yield point"?

Referring to your graph which is for a ductile material I suggest the following. A is the limit of proportionality up to which the stress and strain are proportional to one another and when ...
Farcher's user avatar
  • 97.9k
9 votes
Accepted

How does hot air rise?

The reason for this is actually the temperature-dependent distribution of the absolute value of particle velocities (higher temperature results in higher average velocities as well as greater variance ...
2b-t's user avatar
  • 1,809
9 votes

Is it possible to derive Navier-Stokes equations of fluid mechanics from the Standard Model?

From your comment : So is it possible to prove the consistence of fluid mechanics with the Standard Model? The standard model is consistent with special relativity and quantum theory. We know those ...
StephenG - Help Ukraine's user avatar
9 votes

Differential charge existing

You're mixing up two descriptions that are, in practice, separate. $i=dq/dt$ is usually used in macroscopic physics, when it is understood that you don't study actual individual electrons. In fact, ...
Miyase's user avatar
  • 6,225
8 votes
Accepted

Why does destructive interference not stop a wave?

The state of the rope is defined not just by the vertical displacement along its length, but also by the vertical velocity. In this case, the displacement is zero, but the velocity is not. The parts ...
Buzz's user avatar
  • 16.3k
8 votes
Accepted

Why is there a density instead of mass in the Navier-Stokes Equation, if it's analogue to Newton's Second Law?

The Navier-Stokes equation describes the motion of some infinitesimal volume of the fluid. That is we divide the fluid up into tiny volumes $dV$ and the equation tells us how these tiny volumes move. ...
John Rennie's user avatar

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