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10

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 ...


8

I have at least an answer : Does this depend on the size of the funnel ? Yes, as it was expected (for a very large funnel, all sand falls in the same time, whereas for a very small funnel it doesn't fall at all. More precisely ... the complete answer is probably extremely complex, as one has to take into account the shape of the hourglass, the dynamics ...


7

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 ...


6

Zasso pointed it already out: Scaling up a ant to human size means volume (weight) increasing by length proportional $l^{3}$, but the force of muscles is determined by cross section (not muscle weight), so muscle force goes proportioal to $l^{2}$. Smaller factors are likely: stiffness (or strentgh of the skeleton) balance point (center of mass) leverage ...


6

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 ...


6

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, ...


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

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


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

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

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 ...


5

The OP acknowledges that $m_e/m_p$ is constant and correctly points out that the fine sturcture energy levels of the hydrogen atom are proportional to $m_e$ (as opposed to $m_e/m_p$). It is also true that the energy levels of the Lamb Shift are proportional to $m_e$. However, the hyperfine structure of the hydrogen atom is proportional to $m_e/m_p$. ...


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

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: $$ T = 2\pi \sqrt{\frac{r^3}{GM}} $$ ...


4

Note that by changing the overall scale by a factor k, you are changing the volume of the gasoline by k^3, and the area you are viewing by k^2. So the overall size of the explosion (ie visible flames etc) is not invariant. To find out what is k, you'd have to know lots of details about the fuel etc I suppose.


4

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 ...


4

It is difficult to determine which will fare better. Small mammals can survive a fall from arbitrary distances. Here's one article I found talking about cats. A chief contributor to small mammals' survival is that they have a lower terminal velocity due to the way wind resistance scales. Wind resistance scales with the area of the animal, while weight ...


4

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

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 ...


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 ...


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

This type of equation is not solved, it is piddled with. You have to understand the relations between the values it implies, together with constraints from continuity. When you say about a function that \tilde\psi(s)= g(\tilde\psi(h(s))), you are saying that the value at x is related to the value at h(s). This means that it is related to the value at ...


3

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 $$W_{\mathrm{nc}} = F d = \frac{1}{2}m v^2 + m g d = m g h.$$ ...


2

I think something like this really works.* Here's how mainstream physicists would describe it. Let's use particle-physics natural units ($c=\hbar=1$). Then we have just one dimensionful unit, usually the eV. Nothing is stopping you from using a different unit, the cV (short for "crazy eV"), where the conversion is 1cV = 1eV in this galaxy, but 1cV=0.99eV in ...


2

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 ...


2

What you are asking about is similitude, the linked wikipedia article does a very good job of explaining what you should aim for. You are also going to want to be familiar with Buckingham's $\pi$ theorem... Consider the impact force, $F$. It will clearly depend on some parameters of the model cpasule, say its size, $L$, its mass, $M$, and the velocity of ...


2

Here's Coulomb's Law: If you scale everything by $\lambda$, you get $\frac{1}{\lambda^2}$ in the denominator, but you must also introduce a Jacobian for the integral. For a volume charge, the Jacobian is $\lambda^3$, so you're on the surface of a ball of charge and you make the ball bigger (with the same charge density), the E-field increases ...


2

One way to see this is to just work the math out. Put a test charge Q at a vertical distance $z$ above a surface with a constant surface charge density $\sigma$. The potential clearly depends only on the vertical distance too: $$ \phi(z) = Q\sigma\int_0^{2\pi}\!d\theta \int_0^\infty\!dr\,\frac{r}{\sqrt{r^2+z^2}}\,. $$ Here $r$ is the radial coordinate in a ...


2

A rough idea is this. The strength of your muscles scales as the square of their radius (if you imagine them being like some tubes or something) and your weight scales with the radius squared times height or with the third power (assuming your density is constant during the shrinking process). Then the smaller size will result in less strength but also a lot ...


2

I am taking a different approach to answer this question. I'm terms of numbers there are approximately 8,000,000,000 humans on the planet there are more than 8,000,000,000 star systems in our galaxy alone. There are also just as many if not more smaller systems ie. Molecular systemm. In terms of quantity that easily places humans in the middle/large group. ...



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