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11

The convex hull of the polyhedron will roll on a horizontal plane spontaneously, from being seated on one face to being seated on another, only if (not necessarily if) the centre of mass of the polyhedron is lowered in doing so. Since the convex hull of the polyhedron has only a finite number of faces, there can be only a finite number of such lowerings, and ...


5

Soap films create a surface with minimal surface area naturally. There is some more at this site. Here is a video. Note that these are local rather than global minima.


4

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4

The idea is that the bottle has symmetry about its vertical axis; we are told that every horizontal cross section of the bottle is circular. This means that the shampoo fills it in 'discs' centered about that axis. If a disc has vertical thickness $\Delta h$, then the volume of a disc is $\Delta V = \pi r^2 \Delta h$ where $r$ is the radius. Then as ...


3

However, it seems to me that since $\dot{\mathbf q}=\frac{d\mathbf q}{dt}$, it's only necessary to specify $F(\mathbf q,t)$ for a complete description. While it's true that the function $q$ determines its derivative $\frac{dq}{dt}$, it's not true that the value of $q$ at a particular value $t_0$ of $t$ determines the value of the derivative ...


3

$SO^+(3,1)$ is the so-called restricted Lorentz group, which is the identity component of the Lorentz group $SO(3,1)$. It is a six-dimensional real Lie group, which is not simply connected. Since $SO^+(1,3)$ is not compact, but $SU(2)\times SU(2)$ is compact, the groups cannot be isomorphic as real Lie groups. We have $SO^+(3,1)\simeq ...


3

We can think of units in a formal sense: they can be multiplied and divided, but different units cannot be added. In one way, this is like working in a space of real numbers where each axis represents a power of a dimension, those axes are closed under addition, and that multiplication sends us to a different axis (i.e. you can do meters + meters, but not ...


3

Consider the set of $1\times1$ matrices with real entries. We know how to add these. $[a]+[b]=[a+b]$. We know how to multiply them. [a][b]=[ab]. We can do division, subtraction, exponentiation, etc. It turns out they behave just like the real numbers! It seems that the only difference is that we put some brackets around them. (Turns out they're isomorphic ...


3

By the Fundamental Theorem of Calculus, $$ v(t)-v(t_1)=\int_{t_1}^{t}a(s)\ ds. $$ Similarly, $$ \begin{align*} x(t)-x(t_1)&=\int_{t_1}^t v(s)\ ds\\ &=\int_{t_1}^t\left(v(t_1)+\int_{t_1}^{s}a(u)\ du\right)\ ds\\ &= (t-t_1)v(t_1)+\int_{t_1}^t\int_{t_1}^{s}a(u)\ du\ ds. \end{align*} $$ Rearranging yields the solution $$ ...


3

A parabolic segment looks like the shaded region of this graph: The height of the dam is what the problem says is the "depth", that is, $4$ meters. The face of the dam is the region that is under the line $y=4$ and above the line $y=ax^2$ for some suitable $a$ (so that the bottom edge meets the top edge at $x=6$). If the problem is the least bit ...


2

Here is a portion of the answer. All of these calculations are coming from here. In terms of that website, we think of $SU(3)$ as $A_2$. Then, the $10$ dim rep has heighest weight $(3,0)$ in their notation (and the $\overline{10}$ has heighest weight $(0,3)$). The $27$ d rep is $(2,2)$. The notation the website uses is $X[3,0]$, $X[0,3]$, or $X[2,2]$ ...


2

It is simply a statement about the maximum additive order of something in the ring. Quotients of $\Bbb Z$ are a natural source of rings with different characteristics, of course. The most physical analogy that comes to mind is modular arithmetic. If you're familiar with any sort of cyclic behavior that repeats after finitely many steps, you can view the ...


2

You found the time correctly. Now, in that same time, the boat travels $15$m. So, $$x=vt$$ $$15=v(3.1)$$ You get $v=4.84$m/s. Is that correct?


2

The blue arrow show the two stages of where the camel walked. The red arrow passes from where it wound up to Oasis B. North is at the top of this graph. When the camel was facing South (toward the bottom of the graph), "West of South" would be 15ยบ to the camel's right, since West is to the right when facing South (that is leftward from bottom on the ...


2

I expect it's involved (as well as counterproductive to dispelling your sense of unease) to give a general answer, and impossible to give a non-controversial account. But assuming I've understood your question, here's one way to think of things: There's a mathematical notion of the real number system (a complete ordered field), whose properties are ...


2

It will tip over when the COM is outside the convex hull of the support area. For a polyhedron, the convex hull will be a convex polygon. The COM can be lowered by pivoting around the nearest edge of the convex hull.


2

The notations in physics books are not clear. Assume that $C$ is a constant. If you define $Q(t)$ to be the total heat added to the object, then in fact it should be $\Delta Q=C\Delta T$. Taking derivatives gives $\frac{dQ}{dt}=C\frac{dT}{dt}$. However, in physics books, usually $Q$ is used to mean change in internal energy of an object by heating. So ...


1

Lagrangian mechanics is based on Newtonian mechanics (specifically d'Alembert's principle of virtual works), i.e. sets of second order ODEs. Under some regularity conditions, these second order ODEs can be turned into a system of first order ODEs, where the $q$s and the $\dot q$s are independent variables. Hence if you want to retrieve the equations of ...


1

You are applying the divergence theorem. Your approach looks fine, but I would not say that $$ \frac{dx\, dy\, dz}{\partial x} = dy \, dz, $$ since it is so meaningless that it would deserve a whole theory to make it rigorous. Since $\nabla \cdot E$ is the divergence of $E$, and since $Q=\int \rho\, dx\, dy\, dz$, just apply the divergence theorem.


1

I really wondering why nobody has already mentioned this classic books: Classical Mathematical Physics - W. Thirring Quantum Mathematical Physics - W. Thirring If someone already knows a lot about Topology and Differential Geometry, Chapters 1, 12 and 13 from Geometry, Topology and Physics - M. Nakahara will give some nice introduction to the ...


1

When $n>1$ your expectation $\langle v\rangle$ from L'Hopitals rule, goes to zero as $\tau \to \infty$. (I have not checked your algebra here, but that agrees with my own derivation for the $n>1$ case) You are asked about the average velocity from t=0 untill the particle stops. My reading is that the particle stops after a finite time, in which case ...


1

Let's say that you have two objects colliding, one with momentum $p_1$ and the other momentum $p_2$. You are treating the collision as instantaneous, and so you are applying an impulse $J_i$ (instantaneous change of momentum, i.e. Dirac delta of force) to each body. Conservation of momentum dictates that $J_1 = J, J_2 = -J$ for some $J$; this is the ...


1

Presumably, any solid introduction to smooth manifolds and differential geometry will do. I'm currently lending my copy of Physics for Mathematicians, Mechanics I to a friend, so I can't say for sure. I might recommend fellow Math SE user John Lee's Introduction to Smooth Manifolds. However, I would still recommend Spivak here, especially since he wrote the ...


1

You can view units as being transcendental elements adjoined to the field of real numbers. This is why you can't add two different units in the real-world unless some fundamental relationship is known. For example if we didn't know that energy is somehow related to mass, distance and time, but could measure amount of kinetic energy in some way, it would be ...


1

If you wish, you can regard all physical statements to be only about numbers and units only to be a means to formulate the statements, which do not relate to the mathematics involved. For example, we could reformulate If we weld a wire of length $x$ to a wire of length $y$, the resulting wire has length $x+y$. (1A) such that it does not use any ...


1

Notation such as $a(t)$ is fine and well for some purposes but it tends to make things more complicated when you want to average things over a distance. Assuming the function from time $t$ to distance $x(t)$ is invertible, the object in question has a unique acceleration at each point along its path. You write $a(t(x))$ for the acceleration, but you can ...


1

Hint: You probably realize the camel's first segment puts him somewhere on a circle of radius $23$ around the origin. But in what direction? Well, you're told that it isn't due south, but rather rotated $15^{\circ}$ in a westerly way from due south. Since south is "down", west is "clockwise" along that circle from there. Can you get it from here?


1

There are definitely huge overlaps between math and physics, and the tools you learn in math can be incredibly useful in physics. If you have a heavy math background, then you are uniquely prepared to tackle many theoretical problems in physics that require mathematical sophistication. Relativity is built on differential geometry, high energy physics uses ...


1

I assume from the comments that $P$ is a rank-1 projection, and so it is of the form $$P=|\psi\rangle\langle\psi|$$ for some vector $|\psi\rangle$ in the Hilbert space. Observe that, in a sense, $|\psi\rangle^*=\langle\psi|$ in Dirac notation, whence $$P^* = (\langle\psi|)^*(|\psi\rangle)^*,$$ and since the $*$ is involutive one has ...


1

Any projection operator can be written in the form $$ P = \sum_{j = 1}^r |\psi_j \rangle \langle \psi_j | $$ Where $\psi_1,\dots,\psi_n$ is an orthonormal basis of our Hilbert space. Given $\psi = c_1\psi_1 + \cdots + c_n \psi_n$, we calculate $$ \langle\psi| P = \langle \psi | \left(\sum_{j = 1}^r |\psi_j \rangle \langle \psi_j | \right) = \sum_{j = 1}^r ...



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