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34

For a packing of grains to stay wet up to a height $h$, the gravitational pressure $\rho g h$ needs to be balanced by the capillary pressure $\sigma cos(\theta)/r$. Here, $r$ represents the effective pore radius of the packing, $\theta$ the wetting angle (angle at which the air-water interface meets the sand grains), $\rho$ the water density, $g$ the ...


21

There is nothing wrong with the argument. The mathematics are quite simple and the conclusion is sound - scale cancels out. Let's consider the essence of the question; How does the scale size of an animal affect the absolute height it can jump? Let's assume an on-the-spot spring jump so we exclude a run-up. Now consider an arbitrary animal (let's call it a ...


18

What's wrong is : For a jump of height h one needs energy proportional to $L^3/h$ Taking L as a measure of animal size then we should actually have $$E \approx Mgh \propto L^3h $$ So not divided by $h$ but multiplied by $h$ ! And a little thought would show that dividing by $h$ would make no sense, as it implies you need less energy to jump higher. ...


17

It is mainly a mathematical reason. Extensive quantities grow with system size. If two quantities scale in the same way with a variable (in this case system size), it cancels out in the division. Mini-example: $A$ and $B$ are extensive physical quantities both dependent on $n$. Their ratio is called $C = A / B$. If you scale the system up, $A$ and $B$ grow ...


13

Here is a heuristic. The actual details will depend on the details of what type of rock it is, and materials science and chemistry beyond my pay grade, but this gives what I think should be the general idea. All rocks get wet when you put water on them, the surface gets slick, and the like. When this happens, what you get is the water adhering to the ...


12

This is an example of "scaling laws". Have a look at http://hep.ucsb.edu/courses/ph6b_99/0111299sci-scaling.html - for once Wikipedia doesn't have a good article on the subject. The strength of a muscle is roughly proportional to the area of a cross section through the muscle, so strength is roughly proportional to size squared. That's why I'm a lot ...


10

The fact that animals are all very different aside, it ignores some important facts. The first big problem is that it ignores the fact that an animal scaled up might not be able to stand at all. By the original argument, $F\propto x^2$ and $mg\propto x^3$, where $x$ is some scaling factor. So it should be obvious that, at some point, $mg>F$ and the ...


9

Strength Strength goes like area. Intuitively, the cross sectional area of a muscle counts the number of muscle fibers (actually, myofibrils). Thus, $S\propto A \propto L^2$. But mass goes like volume, $M\propto V\propto L^3$. Therefore strength is proportional to the $2/3$ power of mass, $$S\propto M^{2/3}.$$ This equation expresses the fact that an ...


8

your question, Can one say that all renormalizable theories are scale invariant but the converse, that every scale invariant theory is renormalizable too is not true? has a sharp answer: no, one cannot say so. Renormalizable theories typically have running coupling constants with non-vanishing beta functions. The second part (what you called the '...


8

There's an interesting book by H. Tennekes on the subject of scaling in flying. If you want to go fast and far then the size of your plane scales up, while the speed of sound gives a limit, approached by a Boeing 747. But if you simply want to get off the ground with little effort (what was meant by "easy" in my book), then it is worth while to be small (I ...


7

Let's assume our gravitational potential is zero at our center of mass just before the jump. Our initial mechanical energy is zero. We do nonconservative work to increase our mechanical energy. Then our feet leave the floor and our kinetic energy diminishes until we reach height $h$. We have $$W_{\mathrm{nc}} = F d = \frac{1}{2}m v^2 + m g d = m g h.$$ ...


7

From "Perturbative quantum field theory" Edward Witten (page 446 in volume 1 of "Quantum fields and strings : A course for mathematicians"):


7

The moore's (empirical) "law" states that the number of transistors in a chip increases exponentially (doubles every 2 years). So the question is : is there a hard limit in the number of transistors in a chip? Or, in other words : Are there limits on the size of a chip and on the size of transistors? Indeed there are (almost). The matter is made of atoms, ...


6

You are referring to scaling laws for the energy confinement time ($\tau_{E}$), which is a key performance parameter for a fusion reactor. For example, a stellarator currently has \begin{equation} \tau_{E} \propto \, a^{2.33} B^{0.85}, \end{equation} where $a$ is the minor radius and $B$ is the toroidal magnetic field. This particular scaling is of the Bohm ...


6

What you have shown is that Newton's law is not scale-invariant for a force $F(x,\dot{x},t)$ that is scale-invariant, since you implicitly assumed that $F$ transforms as a scalar under the dilation1. This is kind of a trivial statement: If the l.h.s. of an equation transforms non-trivially and you assume that the r.h.s. transforms trivially, the equation as ...


5

Seems to me your question contains two physics questions which depend on the definition of "easier". Certainly in an atmosphere it is easier to balance gravity the larger the ratio of surface to weight due to the viscosity of the medium. On the other hand this does not make "easier" the maneuverability of the system in energy demands. So you are asking ...


5

I'm not attempting to completely answer your question, but add my 2 cents. When doing condensed matter (statistical) physics, one can see that when a material approaches a (2nd order) phase transition, there will be no natural length scale in the sample (length scale --> infinity). This is the whole idea behind the renormalization group -- you keep "...


5

The best you will get is ‘middle/small’, assuming you treat humans as a whole. Here’s why: Usually, ‘large scale’ physics as given by GR (or even SR) does not apply to us: The gravitational force between two humans is small and the curvature in spacetime caused by a human being is absolutely negligible. At the same time, ‘small scale’ physics, described by ...


5

Suppose that, for a temperature $T_1$, you know $$ \rho(\lambda,T_1) = \lambda^{-5}f(\lambda T_1) $$ for every value of $\lambda$. Now, for a temperature $T_2$, let's introduce a variable $$\bar{\lambda} = \lambda T_2/T_1. $$ Then $$ \begin{align} \rho(\lambda,T_2) &= \lambda^{-5}f(\lambda T_2)\\ &= (T_2/T_1)^5 \,\bar{\lambda}^{-5}f(\bar{\lambda} ...


5

There isn't a simple answer to your question. The scaling will be different in different situations. Let's take your example of gravity. The acceleration is given by: $$ a = G \frac{M}{r^2} $$ so $a$ scales as mass$^1$ and distance$^{-2}$. But consider some other quantity like the orbital period, which is given by: $$ T = 2\pi \sqrt{\frac{r^3}{GM}} $$ ...


5

Wetting here is most likely a capillary effect: Your question is about the size of the air gaps between grains (or rocks) of sand, not the size of the grains or rocks. In practice, except perhaps for very peculiarly shaped objects, these will be of similar magnitude to the smallest gap-filling grains. What happens is that the energy required to create the ...


4

One could make an argument that we are just about the size we need to be. There is a fascinating paper from 1980 by William H. Press: Man's size in terms of fundamental constants, where he argues that intelligent beings have to have a scale of $$ L_H \sim \left( \frac{\hbar^2}{m_e e^2} \right) \left( \frac{ e^2 }{ G m_p^2 } \right)^{1/4} \sim a_0 10^9 \...


4

Think about it like this: in order to get all lengths (i.e. your own length as well as the height of your jump) to scale down by a factor $\alpha$, while keeping the contraction velocity of your muscles the same, you have to rescale all lengths and all times occurring in the problem. That means that you have to scale down the gravity (length over time ...


4

It's because fractal systems are, pretty much by definition, self-similar which means that there is no preferred length scale. If something else depends on the length scale $L$ as a function $f(L)$, the argument $L$ must have units – and no unit is better than any other – so it is "dimensionful". On the other hand, $f(L)$ is a quantity that must have well-...


4

It is the timescale required by viscosity $\nu$ to diffuse momentum significantly over a characteristic length scale $L$. Analogous to how mass and heat can be transfered by molecular diffusion through collisions between particles, momentum can also be transfered through similar mechanisms. For mass and heat transfer the respective time scales are $L^2/\...


4

Assume for now that both P,V are extensive quantities. By the definition of an extensive quantity if the size of system is increased by a factor of λ the extensive quantity is multiplied by λ but the intensive quantities remain the same. So P->(λP) and V->(λV) therefore $$(PV)\rightarrow λ^{2}(PV) $$ This λ square might be the non-linearity that the ...


4

I basically agree with the answer of GiorgioP and only have a little more to add to it. First, statements in words can be just as rigorous as statements in mathematical symbols, because in the end they are two ways of presenting precisely the same assertion. The assertion in terms of maths is that a quantity $A$ is extensive if it is homogeneous of degree ...


3

The difference between the $\mu$-problem and the hierarchy problem is that loop corrections to the value of $\mu$ in MSSM are small and convergent, because of supersymmetry, while the loop corrections to $m_h^2$ in the SM are divergent. So to explain why $\mu$ is small, it is enough to explain why its approximate – tree-level – value is small. (Well, the ...


3

Leaving alone the feathers and everything, I would look at the power law. As I am unaware of the powerlaw concerning fluids (e.g. the interaction with air in this case), I would even ignore it and look at the drivetrain. As most birds take off more like choppers and less like planes (landing on spot) they need most of their muscle for liftof and can glide ...


3

I will give some hints: It is anomalous scaling dimension. scaling dimension is defined as $$x \rightarrow \lambda x,\\ \phi(x) \rightarrow \lambda^\Delta \phi(\lambda x) $$ From the formula (3.45) in the reference (maybe it is better to be $\phi(x) \rightarrow \Lambda^\frac{d-2}{2}\phi(\Lambda^{-1 }x)$), we know that the classical dimension of $\phi$ is $\...


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