Square Definition and 1000 Threads

My best attempt at this question is i have used Q3 as my origin, calculated all the horizontal and vertical forces acting on it due to the other charges, without inlcuding tension, and then I did tan(theta) = vertical forces/horizontal forces, which simplifies to an equation of than(theta) =...

We are given a set of points ##(x_i , y_i)##. If we want to fit a line to these points which has the form of ##y=a_0+a_1x##, we need to do it in a way which minimizes the total error E:$$E = \sum_{i=1}^n (y_i - a_0 - a_1x_i)^2$$So we set ##\frac{\partial E}{\partial a_0} = 0## and...

I can't find a short solution without using calculator.

Hello, This is my first post. For background, I have asked Ask The Physicist who has posted our correspondences and answers here: https://www.askthephysicist.com/ask_phys_q&a.html Search "My question is with regard to an apparent mathematical disagreement" or something like that to find the...

There is a mistake on this textbook (The mistake is pretty obvious) but hey I hope I did not miss something... ought to be ##40425 - 19600##

In which sense(s) do square integrable functions go to zero at infinity? Of course, they cannot go to zero at infinity in the sense of point evaluation, because point evaluation is not the appropriate concept for square integrable functions. There was a recent discussion in the Quantum Physics...

I found a the answer in a script from a couple years ago. It says the kinetic energy is $$ T = \frac{1}{2} m (\dot{\vec{x}}^\prime)^2 = \frac{1}{2} m \left[ \dot{\vec{x}} + \vec{\omega} \times (\vec{a} + \vec{x}) \right]^2 $$ However, it doesn't show the rotation matrix ##R##. This would imply...

Inverse square law would reduce the gravity from the parts of Earth that are farthest from our feet. It'll also reduce the gravity from Earth's center by a lesser amount, but would that be lesser enough so the gravity 20 kilometers under our feet is stronger than the core's gravity or even the...

Many people have said that the noise that affects laser light is proportional to the square root of the illumination. But I can't find the formula. Can anyone help?

Does this mean for x=1 , ##2(1)-1= -\sqrt{2-1}## is false. x=1 is not a solution. But as we square the above equation , ##(2(1)-1)^2=(-\sqrt{2-1})^2## , false equation becomes true. So now x=1 is solution to the new equation ? (Here is the paragraph attached) from book James Stewart.

Hi All, I am working on a project where I am using a donut shaped 1.5 x 1.5 square tubing. It is mild steel and I am assuming it has a 16 gauge wall thickness. I bought this already fabricated and don't know for sure (its possible its 14 gauge). The outside dimension of the donut is 89.5". I am...

I'm getting a quartic in ##c##, is there a way around that...perhaps something elegant I'm missing?

What I am lost about is b, rather the rest of B. I am not sure what it means by probability density and a stationary state.

For this problem, The solution is, I understand their logic for their equation, but when I was trying to solve this problem, I came up with a different expression: ##\Delta A = \Delta L_x\Delta L_y## ##\Delta L_x =\Delta L_y = \Delta L## since this is a square. ##\Delta A = \Delta L^2##...

Hi all I want to calculate the reduced Chi square and root mean square deviation RMSD of some data points that i have, but I am confused about the correct formula for each of them, which one is the correct one. I found this formula in a paper where they referred to it as the RMSD : $$...

For this problem, How do we calculate the moment of inertia of (2) and (3)? For (3) I have tried, ##I_z = \int r^2 \, dm ## ## ds = r ## ##d\theta ## ##\lambda = \frac {dm}{ds}## ##\lambda ## ##ds = dm ## ## \lambda r ## ##d\theta = dm ## ##I_z = \lambda \int r^3 d\theta ## ##I_z = \lambda...

In a lesson on square roots this came up (Root) 27 simplifies too 3(root)3 ok. when I work that out it's = 5.196... or if I say 3squard (root)3 this works out to 15.588.... What am I missing?

I tried solving the problem above by using conservation of energy ##U_{Ei} = U_{Ef} + KE ## ##\frac{4k_eq^2}{\sqrt{2}L} = \frac{4k_eq^2}{2\sqrt{2}L} + 4(\frac{mv^2}{2}) ## ##\frac{2k_eq^2}{\sqrt{2}L} = 2mv^2 ## ## v = \sqrt {\frac {k_eq^2}{\sqrt{2}Lm}} ## However, the solutions solved the...

...y and Coulomb's law diverge as ##r\rightarrow##0? I mean, if a point light source emits light omnidirectionally, the intensity converges at the source, right? THIS is how I should've worded my previous post!

I'm a marine engine mechanic, and as engine controls & sensor systems have gotten more complicated with current technology, my shop gets more & more requests for instrumentation & control system repairs. I have a lot of trouble getting technical info from suppliers, so I have been starting to...

The correct answer is; ##\sqrt{\dfrac{16}{64}}=\dfrac{4}{8}## . I do not seem to understand why some go ahead to simplify ##\dfrac{4}{8}## and getting ##\dfrac{1}{2}## which is clearly wrong. I do not know if any of you are experiencing this... I guess more emphasis on my part. Cheers! Your...

Hey all, I am having trouble solving the following equation for C $$A(-\sqrt{C^2+4F_{+}}-C) = B(\sqrt{C^2+4F_{-}}+C)$$ I don't know how to get ride of the square roots on both sides. Any help would be appreciated, thanks!

$$\lim_{n \rightarrow \infty} \sin^{2} (\pi \sqrt{n^2+n})$$ $$=\lim_{n \rightarrow \infty} \sin^{2} (\pi \sqrt{n^2+n}-n\pi)$$ $$=\lim_{n \rightarrow \infty} \sin^{2} (\pi \sqrt{n^2+n}-n\pi)$$ $$=\lim_{n \rightarrow \infty} \sin^{2} (\pi (\sqrt{n^2+n}-n))$$ $$=\lim_{n \rightarrow \infty} \sin^{2}...

Newton arrived at "there is a force that drives a planet around the star by examining kepler's laws but how did he arrive to inverse square law by kepler's third law (##T^2=\frac {4\pi r^3}{GM}##)? Thank you.

I can do the question using brute force. First I multiply both the numerator and denominator by ##\sqrt{5} + \sqrt{3} - \sqrt{2}## then I simplify everything and rationalize again until no more square root in the denominator. I want to ask if there is a trick to reduce the monstrous calculation...

I placed my Oxy coordinate system at the center of the square, the ##x##-axis pointing rightwards and the ##y##-axis pointing upwards. I divided the square into thin vertical strips, each of height ##h=2(\frac{L}{\sqrt{2}}-x)##, base ##dx## and mass ##dm=\sigma h...

In a) I get that T should be largest where V_0 is least wide, because when V_0 is infinitely wide the particle would be fully reflected. But I don't get how height in b) and energy levels height in c) correlates to T and R. Is it because of their k? I get the opposite answer from the correct...

I have solved c), but don’t know how to solve the integral in d. It looks like an integral to get c_n (photo below), but I still can’t figure out what to make of c) in the integral of d). I also thought maybe you can rewrite c) into an initial wave function (photo below) with A,x,a but don’t...

Proof: Let ## a ## be any integer. Then ## a\equiv 0, 1, 2, 3, 4, 5, 6, 7 ##, or ## 8\pmod {9} ##. This means ## a^{2}\equiv 0, 1, 4, 9, 7, 7, 0, 4 ##, or ## 1\pmod {9} ## and ## a^{3}\equiv 0, 1, 8, 0, 1, 8, 0, 1 ##, or ## 8\pmod {9} ##. Thus ## a^{2}\equiv 0, 1, 4 ##, or ## 7\pmod {9} ## and...

Suppose a square lattice. The planes are such as the image below: I light wave incides perpendicular to the square lattice. The first maximum occurs for bragg angle (angle with the plane (griding angle) as ##\theta_B = 30°## (blue/green), green/blue in the figure). The angle that the...

If I wanted to remove c from the square root, ## r ={ \sqrt{c^2} + {x} } ## would this be correct ## r = \sqrt { {c} + {x} } {c} ## ?

Proof: Let ## n>1 ## be an integer. By definition of factorial, ## n!=n\times (n-1)\times \dotsb \times 1 ##. Now we consider two cases. Case #1: Suppose ## n ## is odd. Then ## n=2k+1 ## for ## k\geq 1 ##. Note that ## n!=(2k+1)!=(2k+1)\times (2k+1-1)\times \dotsb \times 1\implies (2k+1)\times...

[MHB thread moved to the PF schoolwork forums by a PF Mentor] For every square matrix A, C=A(A^t)+(A^t)A is symmetric.

Find the solution here; Find my approach below; In my working i have; ##A_{minor sector}##=##\frac {128.1^0}{360^0}×π×5×5=27.947cm^2## ##A_{triangle}##=##\frac {1}{2}####×5×5×sin 128.1^0=9.8366cm^2## ##A_3##=##\frac {90^0}{360^0}####×π×10×10##=##78.53cm^2## ##A_{major...

Proof: Suppose a positive integer ## a>1 ## is a square. Then we have ## a=b^2 ## for some ## b\in\mathbb{Z} ##, where ## b=p_{1}^{n_{1}} p_{2}^{n_{2}} \dotsb p_{r}^{n_{r}} ## such that each ## n_{i} ## is a positive integer and ## p_{i}'s ## are prime for ## i=1,2,3,...,r ## with ##...

I had a homework question that gives A as an arbitrary matrix. Then the question states that A^2=A Now I manipulate the equation to give this A^2-A=0. -->A(A-I)= 0 So A can be I or 0 Are there any other values A can take?

Proof: Suppose that p is a prime and 3p+1=n^2 for some n##\in\mathbb{Z}##. Then we have 3p+1=n^2 3p=n^2-1 3p=(n+1)(n-1). Since n+1>3 for ##\forall...

When you use the power rule to differentiate the square root, the result is 1/2(sqrt. x) which is undefined at 0. But, when you use the definition of the definition of the derivative to calculate it, the result is infinity. What causes this difference between these two methods?

Hello, I have to solve this problem. I will apply the Niemeijer Van Leeuwen method once I have the probability distribution proper to the renormalization group ,P(s,s'). For example, in the case of a triangular lattice, this distribution is: where I is the block index. However, it is very...

I am trying to numerically solve (with Mathematica) a relativistic version of infinite square well with an oscillating wall using Klein-Gordon equation. Firstly, I transform my spatial coordinate ## x \to y = \frac{x}{L[t]} ## to make the wall look static (this transformation is used a lot in...

I recently encountered this problem on a test where the solution for the above problem was given as follows: $$F= \frac{Gm_1m_2} {r^2} $$ (1) but $$ m=\frac{4}{3}\pi R^3 $$ substituting in equation (1) $$F= \frac{{G(\frac{4}{3}\pi R^3\rho})^2 }{2R^2} $$ where r=radii of the two spheres m=mass...

Hi all There is RF signal in frequency range of 240 MHz to 500 MHz which has been amplitude modulated by 155 Hz square wave signal. The problem is to recover 155 Hz signal while exact RF Carrier frequency is unknown (240 MHz to 500 MHz). Is there any ready made COTs solution available for such...

I'd appreciate if someone could check whether my work is correct. The ##x##-##y## symmetry of the aperture separates the Fresnel integral:\begin{align*} a_p \propto \int_{-a/2}^{a/2} \mathrm{exp}\left(\frac{ikx^2}{2R} \right) dx \int_{-a/2}^{a/2} \mathrm{exp}\left(\frac{iky^2}{2R} \right) dy...

Is it always true that the square root of an odd powered integer will always be irrational?

##\sqrt{3}## is irrational. The negation of the statement is that ##\sqrt{3}## is rational. ##\sqrt{3}## is rational if there exist nonzero integers ##a## and ##b## such that ##\frac{a}{b}=\sqrt 3##. The fundamental theorem of arithmetic states that every integer is representable uniquely as a...

Find area of square ABCD if OQ=OF=6.

def find_next_square(sq): # Return the next square if sq is a square, -1 otherwise sq2=(sq**1/2) xyz=isinstance(sq2, int) if (xyz==True): print("Is perfect square") nextsq=sq+1 print("Next perfect square=",nextsq**2) else: print("Not...

def find_next_square(sq): # Return the next square if sq is a square, -1 otherwise sq2=(sq**1/2) xyz=isinstance(sq2, int) if (xyz==True): print("Is perfect square") nextsq=sq+1 print("Next perfect square=",nextsq**2) else: print("Not perfect...

Hey! :giggle: We consider a double roll of the dice. The random variable X describes the number of pips in the first roll of the dice and Y the maximum of the two numbers. The joint distribution and the marginal distributions are given by the following table Using : For all $a,b\in...

Earlier somebody was telling me about a type of object whose square is zero, and that apparently it has some applications to quantum theory (it wasn't explained very well...). Anyone know what he could have been talking about? And no, it was not "0"... :)