# Tag Info

## Hot answers tagged continuum-mechanics

25

I remember that the question in your title was busted in Mythbusters episode 72. A simple google search also gives many other examples. As for single- vs alternate-direction folding, I'm guessing that the latter would allow for more folds. It is the thickness vs length along a fold that basically tells you if a fold is possible, since there is always going ...

23

Imagine a rock on a rope. As you rotate the rope faster and faster, you need to pull stronger and stronger to provide centripetal force that keeps the stone on the orbit. The increasing tension in the rope would eventually break the it. The very same thing would happen with bar (just replace the rock with the bar's center of mass). And naturally, all of this ...

17

Consider a standard volume of $1\textrm{ m}^3$ of air. This contains on the order of $10^{25}$ molecules of O2 and N2. If you needed to simulate or explain the physics occurring in that volume of air, would you want to model $10^{25}$ molecules and all the interactions between them or, say, 100x100x100 cells based on the Navier-Stokes equations? ...

16

There is a real object with relativistic speed of surface - millisecond pulsar. The swiftest spinning pulsar currently known, spinning 716 times a second. Surface speed of such pulsar with radius 16 km is about $7*10^7$ m/s or 24% speed of light. It is calculated that pulsars would break apart if they spun at a rate of more than 1500 rotations per second....

15

What does it mean? The reason they are conservative or non-conservative has to do with the splitting of the derivatives. Consider the conservative derivative: $$\frac{\partial \rho u}{\partial x}$$ When we discretize this, using a simple numerical derivative just to highlight the point, we get: $$\frac{\partial \rho u}{\partial x} \approx \frac{(\rho ... 13 I don't think this sounds unreasonable as an estimate at all. Let's check it. One designs a building as a compromise between two competing factors: One needs all of the load bearing materials to be well mildly loaded - working in their linear region so that there is no danger of their undergoing plastic (irreversible) deformation, creeping then ... 11 Applying a force in the x-direction might change the shape of the material in the y-direction. The only way to capture such an effect is through a tensor. If you have a general force acting on your body$$ \vec F = (F_x, F_y, F_z)^T$$and you are interested in the reaction of the body by looking at its deformation$$ \vec \epsilon = (\epsilon_x, \...

10

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 the substance in question. For example, for water ($H_2O$), 1 mole equals 18 grams. To get this number, remember that hydrogen ($H$) has an atomic mass of $1$....

9

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, see below). The following picture then illustrates the computation: Take two infinitesimally separated points $x$ and $x+\mathrm{d}x$ and let the wave ...

8

You are observing a hydraulic jump. The Wikipedia article is very good, so I won't try to out-do it. In brief summary, when the water starts running out from the place where it hits the sink, the same flux is spread out over a larger and larger circumference as you move out. This means the flow gets shallower and moves more slowly as you move further ...

7

Are you referring to the exact relativistic equivalent to Navier-Stokes equation or a more general Dissipative Relativistic Hydrodynamics Equation? The "relativistic equivalent to Navier-Stokes equation" would be something like this: There would be an energy momentum tensor with the following form: $T_{\mu\nu} = (e+p)u_\mu u_\nu - p g_{\mu\nu} + \tau_{\mu\... 7 It is a quite famous theorem due to Cauchy. Consider an internal portion$S$of a continuous body$C$. There are two kinds of forces acting on it: Forces proportional to the mass, of the form $$\int_V \mu(x) \vec{f}(x) d^3x\tag{0}$$ where$\vec{f}(x)$is the density of force acting on$x \in V$. And forces acting through the surface$\partial V$, the ... 6 Hooke's Law is frequently used to model multi-dimensional materials because the stress tensor is simple (linear). The full expression can be found on Wikipedia. The simplification for 2D is straight forward (drop any terms with a 3 in the subscript). Note that whether deformation in one dimension affects the others is a property of the material and shows up ... 6 The onset of turbulence in fluids is determined by the Reynolds number $$\mathrm{Re} = \frac{vL}{\nu},$$ where$L$is the characteristic length scale,$v$the characteristic velocity, and$\nu$the viscosity. The onset of turbulence in fluids occurs for$\mathrm{Re}$greater than about 1000 or more, depending on geometry. If we want to see the equivalent ... 6 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 equation is not that simple. Let me try show this. Longitudinal Mode Let us assume that when in equilibrium the string, of density$\mu$, is along with the$x\equiv ...

5

Approximatation: $a(n)$ - Area after $n$ folds. $t(n)$ - Thickness after $n$ folds. $d(n) = k\frac{t(n)}{d(n)}$ - Difficulty after $n$ folds. $a(n+1) = \frac{1}{2} a(n) \rightarrow a(n) = c_1 2^{1-n}$ $t(n+1) = 2 t(n) \rightarrow t(n) = c_2 2^{n-1}$ $d(n) = k4^{n-1}$ Physics: You can't fold an atom. Area of atom $= a(N) = c_1 2^{1-N}$ Solve for $... 5 There are many physical intuitions often presented in various texts on fluid dynamics. I won't mention those here. I will, however, mention that mathematically the passage from a particle point of view to a continuum point of view is still a largely un-resolved problem. (With suitable interpretation, this problem was already posed by Hilbert as his 6th of 23 ... 5 In your calculations you assume that your propeller is a rigid body. You cannot use that assumption, when your speeds are not much smaller than the speed of light. Because "there are no rigid bodies in relativity". 5 Indeed, both the strain tensor $$\epsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right) \tag{1}$$ and the stress tensor $$\sigma_{ij}=2\mu\epsilon_{ij}+\lambda\epsilon_{kk}\delta_{ij} \tag{2}$$ are symmetric by definition. However, bear in mind that these definitions are not always valid;$(1)$assumes ... 5 The momentum flux tensor comes from the momentum equation of Navier-Stokes equations: $$\frac{\partial\left(\rho\mathbf{u}\right)}{\partial t}+\nabla\cdot\mathbf{P}=0\tag{1}$$ Or, using indices (where it is easier to see that$\mathbf{P}$is a rank-2 tensor): $$\frac{\partial\left(\rho u_i\right)}{\partial t}+\frac{\partial\Pi_{ij}}{\partial x_j}=0\tag{2} ... 4 I say no. Assuming all the practicalities work, you can get arbitrarily close to c. But not reach c. You can see this easily from the relativstic formula for kinetic energy: E_k = mc^2(\frac{1}{\sqrt{1-v^2/c^2}}-1) As v approaches c, the energy you need to supply to a particle at the end of the bar tends to infinity. 4 remember that in a three-dimensional description of special relativity the impulse of an object is given by$$\mathbf{p} = \gamma m \mathbf{v}$$with the so-called Lorentz-factor$$\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$$Now, do you think you can accelerate the masses within the slab to a speed greater than light or do you think that something is wrong with ... 4 I walked into my local Mech. E. department and had a chat with a continuum mechanist. Two books he showed me that looked like pretty good answers were: Marcelo Epstein and Marek Elzanowski. Material inhomogeneities and their evolution: a geometric approach. Marcelo Epstein. The geometrical language of continuum mechanics. One thing I like about ... 4 The fabric covering the foam pad has a warp and weave (we can assume.) The fabric can stretch with the warp or the weave but not at a 45 degree angle, called the bias direction in sewing. So, when the ball hits the fabric it causes a wave in the fabric which begins to travel outward like a ripple in water. The wave causes distortion of the fabric as it moves,... 4 When a metal spring is stretched beyond it's elastic limit, the metal begins to undergo some plastic deformation. This is a permanent deformation of metal crystals caused by the creation and motion of crystal lattice dislocations. These processes are partially irreversible and some of the work performed to deform the spring is lost as heat. 4 You show the Euler Equations, reduced forms of Navier-Stokes, which are conservation laws for mass, momentum, and energy along with Stoke's hypothesis. Now, conservative/non-conservative has nothing to do with conservation laws. In conservative form, you can directly integrate the derivatives once on a control volume and conserve the flux quantities ... 4 It provides a convenient graphical means of finding the maximum and minimum shear stress, which are important for determining material failure. You don't absolutely need it, but the graphical interpretation of the circular relationship between normal and shear stress is somewhat convenient. I've read good solid mechanics books that give little if any ... 4 The simple answer to find the average number of atoms/molecules per unit volume is.... N/V (average atoms or molecules/$m^3$) = density ($kg/m^3$) * 1000 / atomic(or molecular) mass *$N_a$where$N_a$is Avogadro's number (~$6 \times 10^{23}$) In general in solid or liquid the distance between the nuclei of atoms is approximately 1 Angstrom =$10^{-10}\$ ...

4

Note that the "law of the lever" is essentially a statement of conservation of energy. If you do work over a long distance with a little force, it can (with a lever) be transformed into a lot of force over a short distance. In the case of a plane, the thrust is used to overcome drag - it has nothing to do with lift (as you point out the plane is not going ...

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