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35

If you're pushing a 10-ton truck and it's not moving, you are not doing any work on the truck because the distance $ds=0$ and the nonzero force $F$ isn't enough for the product $F\cdot ds$ to be nonzero. Your muscles may get tired so you feel that you're "doing something" and "spending energy" but it's not the work done on truck. You're just burning the ...


30

Which year? The sidereal year? The tropical year? The anomalistic year? The calendar year (and whose calendar)? The sidereal year is the average amount of time it takes the Earth to make one complete orbit about the Sun with respect to the fixed stars. The tropical year is the amount of average amount of time between successive spring equinoxes. The ...


29

This is actually a very tricky question, mathematically. Physicists may think this question to be trivial. But it takes me one hour in a math summer school to explain the notion of gapped Hamiltonian. To see why it is tricky, let us consider the following statements. Any physical system have a finite number of degrees of freedom (assuming the universe is ...


26

A (rank 2 contravariant) tensor is a vector of vectors. If you have a vector, it's 3 numbers which point in a certain direction. What that means is that they rotate into each other when you do a rotation of coordinates. So that the 3 vector components $V^i$ transform into $$V'^i = A^i_j V^j$$ under a linear transformation of coordinates. A tensor is a ...


24

Pluto is now classified as a dwarf planet. The main difference between a planet and a dwarf planet has to do with the requirement that a planet clear out the material in and near its orbit. Planets do this, dwarf planets do not. The reclassification was triggered by the discovery of many additional object (the Edgeworth-Kuiper Belt) out beyond the orbit ...


23

A second-order tensor can be represented by a matrix, just as a first-order tensor can be represented by an array. But there is more to the tensor than just its arrangement of components; we also need to include how the array transforms upon a change of basis. So tensor is an n-dimensional array satisfying a particular transformation law. So, yes, a ...


22

Depends on what you mean by 'central force'. If your central force is of the form ${\vec F} = f(r){\hat r}$ (the force points radially inward/outward and its magnitude depends only on the distance from the center), then it is easy to show that $\phi = - \int dr f(r)$ is a potential field for the force and generates the force. This is usually what I see ...


20

Weight is the force with which gravity pulls on a mass. Maybe the simplest way to explain the difference is that on the Moon or on Mars, your weight is reduced because gravity is weaker there, but your mass is still the same.


19

Using the distance between the Sun and the Earth, at least for distances within the Solar system, just gives a better feel for the scales involved. You can't really imagine a distance of, say, 1000000000 kilometers -- or at least I can't. (I deliberately didn't include commas in that number, to illustrate the point.) But using a concrete physical distance ...


18

Officially, no -- but there is a weak case to be made that the Moon orbits the Sun rather than the Earth. If you trace the Moon's path in a Sun-centric frame of reference, that path is completely convex. Quoting this Wikipedia article: Unlike most other moons in the Solar System, the trajectory of the Moon is very similar to that of its planet. The ...


18

The main distinction you want to make is between the Green function and the kernel. (I prefer the terminology "Green function" without the 's. Imagine a different name, say, Feynman. People would definitely say the Feynman function, not the Feynman's function. But I digress...) Start with a differential operator, call it $L$. E.g., in the case of ...


16

Matrices are often first introduced to students to represent linear transformations taking vectors from $\mathbb{R}^n$ and mapping them to vectors in $\mathbb{R}^m$. A given linear transformation may be represented by infinitely many different matrices depending on the basis vectors chosen for $\mathbb{R}^n$ and $\mathbb{R}^m$, and a well-defined ...


16

Charge is a fundamental conserved property of particles. It is, if you like, a measure of how much a particle interacts with electromagnetic fields. A particle with charge can produce and be affected by electromagnetic fields. This is what we mean when we say a particle has charge. Its a simple quantised way to measure the coupling strength of particles with ...


16

The constellations are the 88 internationally recognized stellar groupings in the sky that toghether cover the entire celestial sphere. They typcially correspond to a recognziable pattern and many are named from mythology. However, in modern usage, a specified constellation tecnically referres to the entire region of the sky, not just the recognizable star ...


15

One Celsius (or Kelvin) degree as a temperature difference was defined as 1/100 of the temperature difference between the freezing point of water and boiling point of water. We call these points 0 °C and 100 °C, respectively. The number 100 arose because we're used to numbers that are powers of ten because we use the base-ten system. The Celsius degree is ...


15

It is important to remember that words like this get used in spit-balling sessions and then stick. You have to think of a couple of guy sitting by a black board, coffee in hand saying something like OK, OK! SO that doesn't work. But what if we assume these things come in three flavors and ... It's just a word made up on the spot. That said, I think ...


14

The mass, strictly the inertial mass, relates the acceleration of a body to the applied force via Newton's law: $$ F = ma $$ So if you apply a force of 1 Newton to a mass of 1kg it will accelerate at 1m/s$^2$. This is true whether the object is floating in space or in a gravity field e.g. at the Earth's surface. The weight is the force a body exerts when ...


13

As there wasn't a formal definition of a planet until recently, there still isn't one for a moon. But, a few guidelines: It should be in an orbit which is cleared of other objects, that is, not a bunch of objects in the same or very similar orbit. The typical minimum size for consideration is around a km. There is a smaller class known as moonlets that are ...


13

it turns out this is exactly 273.15°C less the melting temperature of water. Actually, "Kelvin" and "degrees Celsius" are defined such that there are 273.16 degrees between absolute zero and the triple point temperature of water. Degrees Celsius are defined as $K - 273.15$. The freezing point of water is a measured quantity and is not exactly 273.15K ...


12

No, it's not exact, and it's not a definition either. Consider that acceleration has a definition that no one will dispute. It is the time derivative of velocity. More than likely, he had in mind the relativistic generalization of the equation. The more general form of the equality is: $$ F = \frac{dp}{dt} $$ You can easily see how this results in $ma$ ...


11

Not all nonlinear systems are chaotic. However a chaotic system is necessarily nonlinear. There doesn't exists a definition for chaos but using the one given by Strogatz, ref 1: Chaos is aperiodic long-termed behavior in a deterministic system that exhibits sensitive dependence on initial conditions. Like explained in the text: aperiodic long-termed ...


11

Potential is a special case of a more general construction in differential geometry. Let's start abstractly and we'll get to the potentials again at the end. Differential forms The framework of differential forms provides a basis for integration on arbitrary manifold. Differential $p$-forms are totally antisymmetric covariant $p$-tensors. What's special ...


11

It's because the Kelvin scale was and still is defined so that as a measure of temperature difference, one kelvin exactly coincides with one Celsius degree. So the temperature in kelvins was defined as the temperature in Celsius degrees minus $A$ where $A=273.15$ °C is the temperature of the absolute zero, without any additional multiplicative factor. When ...


11

What I've been telling the students in the intro mechanics class I TA is that each force corresponds to some amount of work. The individual contributions of work add up to the net (or total) work, just as the individual forces add up to the net force. So, for instance, when you lift a box up from the floor to a table, there are two forces acting on that ...


10

I don't think the situation you mentioned is possible. You're describing two planets, each of which being in each others L3 points (a lagrangian point is a point in an orbit with special gravitational properties, where an object will remain somewhat stationary relative to the body whose orbit it's in). Even our comparatively tiny spacecraft which sit in ...


10

Indeed most examples of unambiguously labeling chiral states fall back on having another pre-labeled chiral object on hand. For a long time it seemed as though "left" and "right" were entirely interchangeable labels. This symmetry is known as parity. However it turns out there is a way to distinguish left from right in a fundamental way; parity is not ...


10

You must always say with respect to what something is a scalar. If we are given a group $G$, something is called a scalar if it is a member of the trivial representation of that group, i.e. if the (symmetry) group does nothing to it. Nothing more, nothing less. In the most common situation, this means that a scalar is a scalar under the rotation group ...


10

The dictionary definition is wrong. For example, time is a scalar in Newtonian mechanics, and time can be negative. That means that time is not completely specified by its magnitude (absolute value). Other examples include charge, energy, and Celsius temperature. The definition could be improved by cutting "is completely specified by its magnitude" and ...


9

Yes, to some extent. Once you choose which of the electron or positron is to be considered the normal particle, then that fixes your choice for the other leptons, because of neutrino mixing. Similarly, choosing one quark to be the normal particle fixes the choice for the other flavors and colors of quarks. But I can't think of a reason within the standard ...



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