66 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 ...
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26 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 ...
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  • 1,652
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 ...
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22 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 ...
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  • 3,205
22 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 ...
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  • 4,394
21 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 ...
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  • 26.4k
20 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 ...
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  • 855
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 ...
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  • 3,424
14 votes

Why are stresses of continuum systems described via a tensor?

Applying a force in the $x$-direction might change the shape of the material in the $y$-direction. The only way to capture such an effect is through a tensor. If you have a general force acting on ...
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14 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,...
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  • 105k
13 votes
Accepted

Can a building get taller at night?

I don't think this sounds unreasonable as an estimate at all. Let's check it. One designs a building as a compromise between two competing factors: One needs all of the load bearing materials to be ...
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13 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 ...
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  • 1,479
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 ...
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10 votes

How many atoms exist within a continuum body?

The number of atoms (or molecules) in a body is given by Avogadro's constant, or $6.022 \times 10^{23}$ per mole. A mole is the amount of material, in grams, equal to the atomic or molecular mass of ...
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10 votes

Breaking the sound barrier underwater

The reason that the speed of sound is a well-defined quantity is that, for small pertubations, the equations which govern the fluid dynamics can be linearised. In that linearised form, the solution ...
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10 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 ...
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  • 4,087
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 ...
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  • 3,802
9 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 ...
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9 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 ...
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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 ...
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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 ...
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8 votes
Accepted

Why are stresses of continuum systems described via a tensor?

It is a quite famous theorem due to Cauchy. Consider an internal portion $S$ of a continuous body $C$. There are two kinds of forces acting on it: Forces proportional to the mass, of the form $$\...
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8 votes

Why do tall buildings have low resonant frequencies?

We can model the 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 =...
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  • 630
8 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 ...
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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 ...
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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. ...
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7 votes
Accepted

Origin of the major symmetry property of the elasticity tensor

You should start with the strain energy density $\psi$, then define: $$ C_{ijkl} = \frac{\partial^2 \psi}{\partial \epsilon_{ij}\partial \epsilon_{kl}}, $$ and then define $$ \sigma_{ij} = C_{ijkl} \...
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7 votes

Why is it said that standing waves do not transfer energy?

Standing waves are always the result of the interference of two (or more) waves. E.g. in a rope or a string: the interference between a wave and the reflected wave (when the first wave reaches the end ...
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