In geometry, a cube is a threedimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices.
The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations.
The cube is dual to the octahedron. It has cubical or octahedral symmetry.
The cube is the only convex polyhedron whose faces are all squares.
My attempt is approaching this problem like the mass spring model. Considering the buoyancy force as spring force. By doing so, we can have the typical massspring equation
𝑚 (𝑑^2 𝑥)/(𝑑𝑡^2 )+Fbuoyancy = 𝐹_𝑒𝑥𝑡
Then I can assuming the displacement a will be the sinusoidal function...
why is it, the ice cube tray freezes first, at the front of the tray closest to the door.
This should be the warmest place of the refrigerator. The back does not have enough time to send the colder air out to atmosphere.
I think i know the answer but lets us explore this..
There are three charges with +1 μC and −1 μC, are placed at the opposite corners of a cube with edges of length 1 cm, and the distance from P to B is 1cm 2. I labeled them as A, P, and B, which is shown in the diagram below. Since we need to find the magnitude of the charge at point P and the...
I am trying to solve a problem where we have to find the maximum angle before a cube starts sliding when said cube is placed inside a spinning hollow cylinder (the cylinder is placed horizontally). The radius of the cylinder is 0.4 m, the coefficient of static friction between the cube and the...
Without using the Divergence Theorem Evaluate the surface integral $\iint\limits_{\sum} f \cdot d\sigma $ of $f(x,y,z) = xi+ yj + zk , \sum: $ boundary of the solid cube S= $\{(x,y,z) = 0\leq x,y,z \leq 1)\}$
My attempt:
Here we have to use the following definition of surface integral.
Note...
The last three digits of ##x^3## must be solely dependent on the last 3 digits of ##x##. So let ##x=a+10b+100c## for integers ##a,b,c##. Then ##x^3 = a^3 + 30 a^2 b + 300 a b^2 + 300 a^2 c +O(1000)## where of course ##O(1000)## don't affect the last 3 digits. Evidently ##a^3## is the only...
Picture for better understanding.
My answer :
I want to know how to solve this problem without using cylindrical. I mean how can we solve this using cube and its sides.
Thanks.
Summary:: Averaging (a power of) semimajor axis to position ratio wrt to time  celestial mechanics
I evaluated it this far, but i don't know how to change the dt to d theta ... the final solution is
supposedly (1e^2)^(3/2) . Any help will be appreciated.
[Image reinserted with correct...
A) and b) should be useful for solving the initial question.
If the truck is at rest initially, the magnitude of the momentum of the ball becomes ##mv'=MV'  mv##, but this may or may not be less than the magnitude ##mv##, depending on how large ##V'## is. ##V' = \frac{m(v+v')}{M}## in this...
I have tried to understand the solution given in the book which is as pasted below. The solution uses Gauss's Law but makes no mention of which Gaussian surface is used. The diagram that I have used to understand this problem is also given at the end. From the diagram, faces OADG, OABE and OEFG...
Hello everyone, I want to build a rotating display cube (will be mounted on a wall, those cubes that has shops logos ect..), my problem is that I don't know where to start and how to attack the rotating mechanism. I have a 1400Rpm moteur (and it's a 1m per 1m cube that weights around 120kg150kg...
Phi = int of (3xi^ + 4j^).vector dA = 3 int of(xdA)
Now we put x= 3 and we get at last 36 N m^2/C.
I am getting confused why E is a fuction of x. How can that be? How can we represent the E and position x on the same coordinate system. Is it right ? Because we know distance is inversely...
I could solve a similar (rather, a specific case of the above) where the current entered through a
corner and left from the corner opposite to it along the body diagonal of the cube. For this specific case, I was able to easily exploit symmetry to deduce the answer (0). However, I cannot think...
<mentor  Epistemology (how do we know or perceive) is not subject PF supports in the science forums  moved to General Discussion>
Since my previous topic The arrangement of our visual system and the objective truth was closed I will open a new one, less philosophical.
So, just imagine that...
I've looked for this in a hundred places but I keep finding the unit length of the diagonal (root 3). I want to find the angle.
i.e. the angle between the cube's diagonal and its three edges at a vertex (all three angles will be identical, of course).
I guess I hoping to figure it out without...
The proof is given in two steps
1. Prove the lemma.
2. Use lemma to prove result.
%%1Lemma%%
Assume ##a\neq0##. Define ##g:((a+1),a+1)\longrightarrow \mathbb{R}## by ##g(x)=\sqrt[3]{x^2}+\sqrt[3]{xa}+\sqrt[3]{a^2}##. Then ##g## is bounded from below by some positive number ##m##...
As the title implies, I have a cube that was given to me new around 19801981. All the info I have found online indicates my cube is likely not an original. I would have thought at that time there wouldn't have been knock offs, but it's entirely possible I am wrong.

The colors on my cube do...
Hey! :giggle:
Let $\displaystyle{W:=\left \{\begin{pmatrix}x\\ y\\ z\end{pmatrix}\in \mathbb{R}^3\mid x,y,z\in \{1,1\}\right \}}$.
Draw the set $W$ in a coordinate system. Let $v=\neq w$ and $v,w\in W$. If they differ only at one coordinate connect these points by a line.
With this...
But I actually don't get the same matrix. What I get is the transpose of the other when I change the order
i.e when I do [A]^2[A] I get the transpose of [A][A]^2 and vice versa
What I'm trying to do is find the cube of the expectation value of x in the harmonic oscillator in matrix form.
We're...
Firstly, the cube root of 17 is 2.571281591 which is 2.57 to 3.s.f.
Initially, I thought about just approaching this problem using the NewtonRaphson Method when x0=2. In which case; x^3=17
x^317=0
Using the NewtonRaphson iterative formula xn+1=xrf(xn)/f’(xn)
f(x)=x^317
f’(x)=3x^2...
I am not sure about value of depth I need to use. What I did:
Hydrostatic pressure at bottom of cube = hydrostatic pressure by oil + hydrostatic pressure by water
= ρoil . g . d + ρwater . g . h
Is it correct I use value d = 0.2 m for depth of oil and value of h = 0.02 m for depth of water to...
Positive integers are written on all the faces of a cube, one on each. At each corner (vertex) of the cube, the product of the numbers on the faces that meet at the corner is written. The sum of the numbers written at all the corners is 2004. If $T$ denotes the sum of the numbers on all the...
Find point d on the line of r(t)=(0,0,0)+(−1,1,1)t which make the triangular pyramid abcd has the volume of 4 unit cube when a(0,0,0),b(1,0,1),c(0,1,0) are the points on the plane of −x+z=0.
Extending the previous problem of Cubes Within Cubes, where we asked how many cubes can be drawn within a cube, now we ask how many cuboids of size {##i ,j ,k = 1,n##} can be drawn in a cube of dimension ##n## where ##n## is an integer.
Dear Recreational Geometry People,
I recovered a thing I did very long ago from a drawing of mine that I fortunately just found again. With some effort I was able reconstruct what I did and redraw it. It is a geometric dissection. The task is to slice up a square and use those pieces to make a...
For convenience, I take the center of the upper face.
The charges at the top cancel each other's effects, and those at the bottom cancel each other's horizontal effects, so I get$$E=4k\frac{q}{r^2}\sin\theta$$I have found that ##\theta=\arcsin\left(\frac{s}{r}\right)=\arcsin{\sqrt\frac{2}{3}}##...
The correct answer is B, but I am not sure why.
I have a few confusions regarding this problem. First of all, I had thought that we cannot use Gauss' Law to determine the flux through a SIDE of a cube since Gauss' Law only works for SURFACES. How can we determine how an electric field pierces a...
So, I recognise that:
$$ma=pg\left( L^{2}\right) \left( Ly\right) \dfrac {1}{2}pgL^{3} $$
whereby $$pg\left( L^{2}\right) \left( Ly\right)$$ is the upthrust while the other is mg.
So, to find the largest speed of cube, I will assume that acceleration is zero since the acceleration slowly...
Homework Statement: Verify the claim of Section 7.2 that the electrons of a metal collide with the surface at a rate of about 10^30 per second per square centimeter. Do this by estimating the collision frequency of electrons in a 1.00cm cube of copper metal with one face of the cube surface...
Summary: Apparently an ice cube gains mass when it melts
So I'm asked to "Find the fractional increase in inertial mass when an ice cube melts ".
All I've got off the top of my head right now is that a cube has energy = mc^2, and then when the cube melts, energy Q = (Heat of fusion)(m) is...
On the right paragraph it says "The trial divisor 1200 goes into the dividend 13952, 8 times"
Clearly 1200 goes into 13952, 11 times. I don't understand why 8 is (arbitrarily?) chosen. Please help. Thanks.
I'm having a little trouble understanding how to go about solving this problem. I was in class Tuesday and the hint I got from the T.A. running my discussion section was that : "because the electric field is only nonuniform along the x axis, the electric field will both enter(negative flux) and...
Given a cube ABCD.EFGH whose side length is 4 cm. If the point I, K, and J is dividing EF, FG, and BG to two equal lengths respectively, determine the volume of pyramid D.IJK!
I think I can work out the pyramid's base area by deriving for the formula of equilateral triangle area. What I can't...
Tried to use gauss law but there isn't any usefull symmetry that I have seen. Also tried to integrate the field due to small charges over the whole cube, didn't work too since the integral were too much complicated.
Homework Statement
Find the center of mass of a homogeneous solid cube with side ##L## analytically.
Homework Equations
None.
The Attempt at a Solution
I don't understand how to find the center of mass on three dimensions. I know that since it is homogeneous, if I center the cube on the...
Homework Statement
Let X = ##\mathbb{R^m}## and . be a Norm on X. The dual norm is defined as ##y_*:=sup({\langle\,x,y\rangle :x \leq 1})##
a) Show that ##._*## is also a norm
b) Construct two norms ##.^O## and ##.^C## so that:
{##x:x^O=1##} is a regular octahedron...
Homework Statement
I have a value of $$ U=U_0+x (∂U/∂x)+y(∂U/∂y)+z (∂U/∂z)+1/2x^2(∂^2U/∂x^2)+1/2y^(2∂^2U/∂y^2)+...$$
We need to find the mean value of the U. So the answer is
$$\overline{\rm U}\approx U_0+a^2/24(∇^2U)$$Homework Equations
$$\overline{\rm U}=1/a^3 \int \int\int Udxdydz$$
The...
Homework Statement
The figure shows a closed Gaussian surface in the shape of a cube of edge length 3.20 m. It lies in a region where the electric field is given by E=(1.65x+2.86)i + 4.39j + 5.47k N/C, with x in meters. What is the net charge contained by the cube?
Homework Equations
net...
Homework Statement
A point charge q is placed inside a cube of side 2a. What will be the flux associated with the lower surface ABCD?
Homework Equations
I think I can apply Gauss Law here, but can't think of something connecting it with the lower surface.
∫B.dl = 1/ε° X Charge Enclosed
The...